LMAM Lin Shu
email: lshu@math.pku.edu.cn

THE REGULARITY OF THE LINEAR DRIFT IN NEGATIVELY CURVED SPACES

Keywords: 2010 Mathematics Subject Classification. 37D40, 58J65 entropy, heat kernel, linear drift, locally symmetric space

We show the linear drift of the Brownian motion on the universal cover of a closed connected Riemannian manifold is C k´2 differentiable along any C k curve in the manifold of C k metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is C 1 differentiable along any C 3 curve of C metrics with negative sectional curvature. We formulate the first derivatives of the linear drift and entropy, respectively, and show they are critical at locally symmetric metrics. y 5.3. The existence of F s y 5.4. Quasi-invariance property of F s y 5.5. The extended map F s 5.6. The differential of λ Þ Ñ p λ pT, x, ¨q 6. Higher order regularity of the heat kernels in metrics 6.1. A sketch of the proof for Theorem 1.3 with i ě 2 6.2. Proofs of the properties concerning φ j λ 7. Regularity of the entropy References

Introduction and statement of results

If we think of curvature as a measure of the geometric complexity of a closed connected Riemannian manifold, the 'simplest' geometric objects are those with constant sectional curvatures since their universal covers must be spheres, planes or Poincaré disks. A little more 'complicated' objects are locally symmetric spaces, whose universal covers are symmetric. An attractive problem in geometry is to characterize locally symmetric spaces using other complexities, for instance, Lichnerowicz's conjecture in 1944 ([Li]) says that symmetry is equivalent to the harmonic property of the space, which means the geodesic spheres have constant mean curvature depending only on their radii.

From the point of view of dynamical systems, geometry influences dynamics and hence the geometric complexities can be read using dynamical complexities. One example is volume entropy, which is the exponential growth rate of the volume of a ball in the universal cover as a function of the radius. It is named entropy since it is no bigger than the topological entropy of the geodesic flow in the unit bundle, with equality if the underlying space is of nonpositive curvature ( [Man]) or the underlying space has no conjugate points and its Riemannian metric is Hölder C 3 ( [FM]). In 1983, Gromov ([Gro]) conjectured that among all metrics of volume equal to the volume of a locally symmetric metric g 0 , the volume entropy is minimized at metrics isometric to g 0 . For negatively curved spaces, this was shown by Katok ([K1]) for the 2-dimensional case and was shown for higher dimensional cases by Besson, Courtois and Gallot ([BCG]). The remarkable rigidity result in [BCG] implies the Mostow rigidity ( [Mos]) (and its generalizations by Corlette ([Cor]), Siu ([Si]) and Thurston ([Th])) and also has many interesting rigidity applications in dynamics combined with the results of [BFL], [FL], [L2], etc. This helps us to understand the interaction between differential geometry and dynamical systems, and leads to many more rigidity studies on both sides.

Since the geodesic flow in the unit tangent bundle always preserves the Liouville measure, its entropy is another natural quantity (besides the topological entropy) for the description of the dynamical complexity of the system. Clearly, the entropy of the Liouville measure is always less or equal to the topological entropy for the geodesic flow. It was conjectured by Katok in 1982 ([K1], see also [BK]) that in the negatively curved manifold case, these two entropies coincide (if and) only if the manifold is locally symmetric. This is true in the 2-dimensional case ([K1]). For the higher dimensional cases, it is a very difficult problem and it depends on our understanding of the dedicate difference between the Liouville measure and the Bowen-Margulis measure (for topological entropy). To approach this conjecture, many experts tried to study the variations of the two entropies with respect to perturbations of the original system and to derive formulas for their infinitesimal changes (see e.g., [Con, Fl, KKPW, KKW, KW, Kn]). We mention some of them briefly. The smoothness of the topological entropy for perturbations of the Anosov flows were considered by Katok, Knieper, Pollicott and Weiss in [KKPW] (see [KKW] for the first order derivative formula of the topological entropy of the geodesic flow under one-parameter family of C 2 perturbations of the original C 2 negative curved metric). (As a corollary of the results of [BCG] and [KKPW], a locally symmetric negatively curved metric g 0 is a critical point of the topological entropy. Whether the reverse is true or not is an open question that was addressed in [KKW].) Contreras ([Con]) continued to analyze the regularity of the Liouville entropy with respect to perturbations of the system. Furthermore, Flaminio ([Fl]) gave a partial positive answer to Katok's conjecture by showing that along any non-trivial deformation the topological entropy and the difference between the topological entropy and the Liouville entropy are locally strictly convex functions of the deformation parameter. Besides its connection with the above rigidity problems, the studies of the regularities of the entropies have their own interest in the dynamical dimension theory (see e.g., [START_REF] Anosov | Geodesic flows on closed Riemann manifolds with negative curvature[END_REF][START_REF] Katok | Nonuniform hyperbolicity and structure of smooth dynamical systems[END_REF][START_REF] Misiurewicz | On non-continuity of topological entropy[END_REF][START_REF] Misiurewicz | Diffeomorphisms without any measure with maximal entropy[END_REF][START_REF] Newhouse | Continuity properties of entropy[END_REF][START_REF] Pollicott | Analyticity of dimensions for hyperbolic surface diffeomorphisms[END_REF][START_REF] Rugh | On the dimensions of conformal repellers. Randomness and parameter dependency[END_REF][START_REF] Yomdin | Volume growth and entropy[END_REF][START_REF] Yomdin | C k -resolution of semialgebraic mappings. Addendum to: "Volume growth and entropy[END_REF]). They are also in the same flavor of the studies of the linear response problems in statistical mechanics for the understanding of the heat conduction (see Ruelle ([Ru1,[START_REF] Ruelle | General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium[END_REF][START_REF] Ruelle | Differentiating the absolutely continuous invariant measure of an interval map f with respect to f[END_REF][START_REF] Ruelle | Differentiation of SRB states for hyperbolic flows[END_REF])). The key step in the linear response theory is to justify, derive and understand the first order derivative of the measure theoretical entropy of the SRB measure under smooth perturbations of the original system (see [START_REF] Ruelle | A review of linear response theory for general differentiable dynamical systems[END_REF] and [B] for nice introductions to this field and hot references). Now, if we consider Brownian motion instead of the geodesic flow, can we find similar connections between the stochastic dynamics and the geometric complexities?

Let M be an m-dimensional orientable closed connected smooth manifold with fundamental group G. Its universal cover space Ă M is such that M " Ă M {G. For a C 2 Riemannian metric g on M , let r g be its G-invariant extension to Ă M . Consider the Brownian motion on p Ă M , r gq with starting point x P Ă M . Its density function of the distribution at time t P R `, denoted by ppt, x, yq, y P Ă M , is the fundamental solution to the heat equation Bu{Bt " ∆u, where ∆ :" Div∇ is the Laplacian of metric r g on C 2 functions on Ă M . Denote by Vol which was introduced by Guivarc'h ( [Gu]). It tells the average in time of the shift of the Brownian motion from its starting point. The other is the (stochastic) entropy h :" lim tÑ`8 ´1 t ż ln ppt, x, yqppt, x, yq dVol r g pyq, which was introduced by Kaimanovich ([Kai1]). It tells the average decay rate of the transition probabilities of the Brownian motion. Both and h are independent of the choice of x and are well-defined since we have a compact quotient.

The linear drift, the stochastic entropy and the volume entropy (denoted by υ) are interrelated as follows:

(1.1)

2 paq ď h pbq ď υ pcq ď υ 2 .
(For paq, see [START_REF] Kaimanovich | Brownian motion and harmonic functions on covering manifolds. An entropic approach[END_REF] for the negatively curved case and see [L4] for the general case. For pbq, see [Gu]. Inequality pcq was derived in [L4] as a corollary of paq and pbq.) All the equalities in (1.1) turn out to be related to the rigidity problem of locally symmetric spaces. The equality 2 " h (and hence υ 2 " h and " υ) implies the space is locally symmetric in the negative curvature case by results in [START_REF] Kaimanovich | Brownian motion and harmonic functions on covering manifolds. An entropic approach[END_REF][START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF][START_REF] Benoist | Flots d'Anosov à distributions stables et instables différentiables[END_REF]FL,[START_REF] Freire | On the entropy of the geodesic flow in manifolds without conjugate points[END_REF] and this characterization continues to hold in the non focal point case [START_REF] Ledrappier | Entropy rigidity of symmetric spaces without focal points[END_REF]).

For h " υ, whether it holds only for locally symmetric spaces is equivalent to a conjecture of Sullivan (see [L2] for a discussion), which is not even known for negatively curved manifolds with dimensions greater than 2. Note that for Brownian motion, it is associated with a natural important probability measure in the unit tangent bundle of M , the so-called harmonic measure (see Section 2.3) and, in the negatively curved case, the quotient h{ is proportional to the Hausdorff dimension of the harmonic measure at the infinity boundary ( [L1]). Hence Sullivan's conjecture depends on the understanding of the dedicate difference between the harmonic measure and the Bowen-Margulis measure. This, together with the works that we mentioned above on Katok's conjecture and the linear response theory, motivates our study in [START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF] and the present paper to analyze the regularities of the linear drift and the stochastic entropy with respect to metric changes, to derive formulas for the corresponding differentials and to understand the critical points.

We need some notations to state our regularity results in a precise form.

For k P N, let C k pS 2 T ˚q be the collection of C k sections of S 2 T ˚, the bundle of symmetric 2-forms on the tangent space T M . It is a Banach space with the topology of the uniform convergence in k derivatives. The set of all smooth sections of S 2 T ˚, denoted by C 8 pS 2 T ˚q :" Ş 8 k"0 C k pS 2 T ˚q, is a Fréchet space whose topology is given by all the C k -norms. Let M k pM q denote the set of C k Riemannian metrics on M . It is the collection of elements in C k pS 2 T ˚q which induces a positive definite inner product on each tangent space T x M , x P M . The space of all smooth Riemannian metrics M 8 pM q " Ş 8 k"1 M k pM q consists of an open convex positive cone in C 8 pS 2 T ˚q and is a Fréchet manifold.

Let k pM q pk ě 3 or k " 8q be the submanifold of M k pM q made of negatively curved C k metrics on M . It is open in M k pM q. For any curve λ P p´1, 1q Þ Ñ g λ P k pM q, the linear drift for each pM, g λ q, denoted by λ , is positive [START_REF] Kaimanovich | Brownian motion and harmonic functions on covering manifolds. An entropic approach[END_REF]Theorem 10]).

Our main result in this paper is the following.

Theorem 1.1. Let M be a closed connected smooth manifold. For any C k pk ě 3q curve λ P p´1, 1q Þ Ñ g λ P k pM q, the function λ Þ Ñ λ is C k´2 differentiable; for any C 8 curve λ P p´1, 1q Þ Ñ g λ P 8 pM q, the function λ Þ Ñ λ is C 8 differentiable.

A special case of Theorem 1.1 was treated in [START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF], where we considered the case that g λ " e 2ϕ λ g is a C 3 curve of C 3 conformal changes of g in 3 pM q and showed the differentiability of λ in λ. In that setting, the relation between the r g λ -Laplacian ∆ λ and the r g-Laplacian ∆ can be formulated as: for f a C 2 function on Ă M , ∆ λ f " e ´2ϕ λ ´∆f `pm ´2qx∇ϕ λ , ∇f y g ¯,

where we still denote ϕ λ its G-invariant extension to Ă M . So we can split the difference λ ´ 0 into two parts corresponding to the time change and drift change of the diffusion, respectively. The first part differentiability can be handled using the results in [FF, LMM], while the second part differentiability was shown in the process of the diffusion using the Cameron-Martin-Girsanov formula and the Central Limit Theorem for the linear drift ( [L3]). There is no such simple picture for the C 1 regularity of the linear drift for general deformation of metrics or for the higher order regularities consideration.

Our strategy to prove Theorem 1.1 is to use the expression of the linear drift at the infinity boundary of Ă M and prove the C k´2 regularity of the ingredients in that formula.

Let r g λ be the G-invariant extensions of g λ in Ă M . The geometric boundary of p Ă M , r g λ q, denoted B Ă M λ , is the collection of the equivalence classes of unit speed r g λ -geodesics that remain a bounded distance apart. Each B Ă M λ can be identified with B Ă M 0 (or simply B Ă M ) since the identity isomorphism from G to itself induces a natural homeomorphism between the two boundaries. For x P Ă M and ξ P B Ă M , let X λ px, ξq be the initial speed vector of the unit speed r g λ -geodesic starting from x belonging to the equivalent class of ξ. Let Div λ be the divergence operator of p Ă M , r g λ q. It is true (see Section 2 for a more precise statement) that

(1.2) λ " ´żM 0 ˆB Ă M Div λ X λ d r m λ ,
where M 0 is a connected fundamental domain and d r m λ " dx λ ˆd r m λ x , where dx λ is proportional dVol r g λ and r m λ x is the hitting probability at B Ă M of the r g λ -Brownian motion starting at x.

The term ´Div λ X λ in (1.2) has its geometric feature as being the mean curvature of the strong stable horosphere of the geodesic flow in the metric r g λ (see (2.4)); its regularity in λ can be deduced using the results from [Con, KKPW, LMM] on the Morse correspondence map between the geodesic flows of two negatively curved spaces (Proposition 3.5).

To conclude Theorem 1.1, we show the following on the regularity in λ of the harmonic measure m λ :" r m λ | SM , where SM :" M 0 ˆB Ă M (see Section 3 for precise definitions).

Theorem 1.2. Let M be a closed connected smooth manifold. For any g P k pM q, k ě 3, there exist a neighborhood V g of g in k pM q and a Banach subspace H 0 b of continuous functions on SM such that for any C k curve λ P p´1, 1q Þ Ñ g λ P V g with g 0 " g, the mapping λ Þ Ñ m λ is C k´2 in the weak topology of the dual space pH 0 b q ˚.

The regularity problem in Theorem 1.2 was not discussed in [START_REF] Ledrappier | Entropy rigidity of symmetric spaces without focal points[END_REF] for the conformal change case. It is subtle since harmonic measures are not the dual of linear functionals acting on the space of continuous functions on SM . For each g λ , it is defined naturally a one parameter family of actions Q λ t pt ě 0q on continuous functions f on SM :

(1.3) Q λ t pf qpx, ξq :" ż M 0 ˆB Ă M r f py, ηqq λ pt, px, ξq, dpy, ηqq, where q λ denotes the transition probability of the g λ -Brownian motion on the stable leaves of SM and r f denotes the G-invariant extension of f to Ă M ˆB Ă M . Since pM, g λ q is negatively curved, it is known ( [L3]) that each Q λ T (for T large) is a contraction on some Banach space H λ b of continuous functions on SM which are Hölder continuous with respect to direction changes and this makes m λ a fixed point of the dual of Q λ T ˇˇH λ b . The idea to prove Theorem 1.2 is to use the classical perturbation result on a linear contraction in a Banach space ( [Kat]). Hence, it suffices to find a common Banach space H 0 b and a T ą 0 such that ' all Q λ T , λ P p´1, 1q, are contractions on H 0 b , uniformly in λ, and

' λ Þ Ñ Q λ T is C k´2 as maps from H 0 b into itself.
To achieve this, we not only need the regularity of the heat kernels q λ in g λ , but also need the estimations on its differentials, which we present with full generality as follows.

For each C k Riemannian metric g " pg ij pxqq P M k pM q, set }g} C a (a ď k) for the C a -norm of g which involves the bounds of tg ij pxqu and of their differentials up to the a-th order. Each C k curve λ P p´1, 1q Þ Ñ g λ P M k pM q defines a one parameter family of tangent vectors X λ " pX λ ij pxqq P C k pS 2 T ˚q. Let pX λ q p0q " X λ , pX λ q plq " `pX λ q pl´1q ˘p1q λ , l " 1, ¨¨¨, k ´1. All pX λ q plq are elements in C k pS 2 T ˚q. By }pX λ q plq } C a (a ď k), we mean the C a -norm of pX λ q plq , which involves the bounds of the pX λ q plq ij pxq and of their differentials in x up to the a-th order.

Let C k,ι p Ă M q denote the collection of C k functions on Ă M with Hölder exponent ι. The set of continuous functions on Ă M is denoted by Cp Ă M q. For any one parameter family of real functions on Ă M or real numbers λ Þ Ñ a λ , let pa λ q piq λ denote the i-th differential in λ whenever it exists.

Theorem 1.3. For any g P M k pM q, k ě 3, there exist ι P p0, 1q and a neighborhood V g of g in M k pM q such that for any C k curve λ P p´1, 1q Þ Ñ g λ P V g with g 0 " g:

i) The mappings λ Þ Ñ p λ pT, x, ¨q, x P Ă M , T P R `, are C k´2 in C k,ι p Ă M q. ii) Let T 0 ą 0 and q ě 1. For each i, 1 ď i ď k ´2, l, 0 ď l ď k ´2 ´i, and T ą T 0 , there exists c λ,pl,iq pqq depending on pl, iq, m, q, T, T 0 , }g λ } C l`i`2 and t}pX λ q pjq } C l`i´j`1 u jďi´1 such that › › ›∇ plq pln p λ q piq λ pT, x, ¨q› › › L q ď c λ,pl,iq pqq,

(1.4)

where the L q -norm is taken with respect to the distribution at T of the r g λ -Brownian motion probability. iii) Let T 0 ą 0 and q ě 1. For each i, 1 ď i ď k ´2, and T ą T 0 , there exists c λ,piq pqq depending on i, m, q, T, T 0 , }g λ } C i`2 and t}pX λ q pjq } C i´j`1 u jďi´1 such that

(1.5)

› › › › › pp λ q piq λ pT, x, ¨q p λ pT, x, ¨q › › › › › L q
ď c λ,piq pqq.

iv) Let r f P Cp Ă M q be uniformly continuous and bounded. Then for any T ą 0 and i, 1 ď i ď k ´2, the function ş Ă M pp λ q piq λ pT, x, yq r f pyq dVol r g λ pyq belongs to Cp Ă M q.

A priori, the derivative in λ of p λ pt, x, yq, if it exists, satisfies the equation

" B
Bt qpt, x, yq " ∆ λ y qpt, x, yq `p∆ λ y q p1q λ p λ pt, x, yq, qp0, x, yq " 0. (1.6) Equation (1.6) always has a solution in the distribution sense. Our Theorem 1.3 is that this distribution is given by a function pp λ q p1q λ pt, x, ¨q P C k,ι p Ă M q and that its gradients satisfy (1.4). This does not follow directly from (1.6) since p∆ λ y q p1q λ p λ pt, x, yq has singularities as t goes to zero and y " x. This type of singularities was not handled in the literature and this difficulty accumulates when we consider tpp λ q piq λ pT, x, ¨qu iě2 . Moreover the universal cover is non-compact. We are not successful to give a more direct proof after trying many classical analysis methods such as parametrix, parabolic Schauder theory, Sobolev spaces, etc. (cf. [Fr, MM, Ro]).

To get an explicit expression of the solution, we use the stochastic calculus representations of the heat kernel and the Brownian motion. Namely, we find a C 1 vector field z λ,1 T pyq on Ă M (see (5.15)) such that, for any smooth f on Ă M with compact support, ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙p1q λ " ż Ă M @ ∇ λ y f pyq, p λ pT, x, yqz λ,1 T pyq D λ dVol λ pyq.

So, using the classical integration by parts formula, we obtain ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙p1q λ " ´ż Ă M f pyq ´Div λ z λ,1 T pyq `@z λ,1 T pyq, ∇ λ ln p λ pT, x, yq D λ ¯pλ pT, x, yq dVol λ pyq. (1.7)

In the same way, we will find C 1 vector fields tz λ,j T pyqu jďiďk´2 (see (6.6) and (6.12)), which will enter the formulas of pln p λ q piq λ and the gradients ∇ plq pln p λ q piq λ . It is not hard to obtain a stochastic expression for z λ,1 T using the Eells-Elworthy-Malliavin construction of the Brownian motion on a manifold. But the associated stochastic differential equation of the Brownian motion in the orthogonal frame bundle is degenerate. So the main technical difficulty is that the C 1 regularity of z λ,1 T does not follow directly from the stochastic pathwise integration by parts theory or the stochastic functional methods for the calculus of variations (cf. [START_REF] Bismut | Martingales, the Malliavin calculus and Hörmander's theorem[END_REF][START_REF] Bismut | Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions[END_REF][START_REF] Driver | A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold[END_REF][START_REF] Driver | A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold[END_REF][START_REF] Malliavin | Stochastic calculus of variations and hypoelliptic operators[END_REF][START_REF] Malliavin | C k -hypoellipticity with degeneracy[END_REF][START_REF] Malliavin | C k -hypoellipticity with degeneracy. II, Stochastic analysis[END_REF][START_REF] Watanabe | Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels[END_REF]). Similar difficulties will also arise in obtaining the stochastic expressions of tz λ,j T pyqu 2ďjăiďk´2 and in using these expressions to identify z λ,i T . However, since we are mainly interested in the behaviors of the projections of the various stochastic objects on the manifold, we can overcome these difficulties by a constructive method using some ideas from [START_REF] Carverhill | Lyapunov exponents for a stochastic analogue of the geodesic flow[END_REF][START_REF] Driver | Integration by parts for heat kernel measures revisited[END_REF][START_REF] Hsu | Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold[END_REF][START_REF] Malliavin | Stochastic Jacobi fields, Partial differential equations and geometry[END_REF]. Most computations to guarantee the constructions will appear in Chapter 4 for the neatness of the paper. It is also for the introduction of the beautiful ideas from [START_REF] Carverhill | Lyapunov exponents for a stochastic analogue of the geodesic flow[END_REF][START_REF] Malliavin | Stochastic Jacobi fields, Partial differential equations and geometry[END_REF] to treat the Brownian motion as a stochastic analogue of the geodesic flow (see Section 4.2 for details). This dynamical point of view will be very helpful in understanding our constructive proof concerning the smoothness of all the vector fields tz λ,j T pyqu 1ďjďk´2 . Note that the stochastic flow (for the Brownian motion) always preserves the Liouville measure ( [CE]). In analogy with Katok's conjecture, one interesting question is when will the entropy of the Liouville measure be equal to the topological entropy for this flow?

In showing Theorem 1.1, we also obtain the formula (3.12) (see the formula (3.13) for a more precise form) for the first order differential of the linear drift under one-parameter family of deformations of negative curved metrics, which implies the following two theorems.

Theorem 1.4. (see Corollary 3.10) Let M be a closed connected smooth manifold. Let g P 3 pM q be a negatively curved locally symmetric metric. Then for any C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q with g 0 " g and constant volume, p λ q 1 0 :" pd λ {dλq| λ"0 " 0. Theorem 1.5. (see Theorem 3.11) There is a linear functional L on C k pS 2 T ˚q such that for all C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q with g 0 " g and constant volume, p λ q 1 0 " LpX q.

A similar approach yields the first order differentiability in λ of the stochastic entropy h λ of the Brownian motion on p Ă M , r g λ q.

Theorem 1.6. Let M be a closed connected smooth manifold. For any C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q, the function λ Þ Ñ h λ is C 1 differentiable and is critical at λ " 0 when g 0 is locally symmetric. Moreover, there is a linear functional K on C k pS 2 T ˚q such that ph λ q 1 0 :" pdh λ {dλq| λ"0 " KpX q.

An explicit formula of KpX q is given in Theorem 7.3, where the infinitesimals of the metric changes appear in a neat way. Hence an interesting question is to characterize the critical points of the entropies of harmonic measures. In our approach, the higher order regularity of λ Þ Ñ h λ and the analysis on the differentials would depend on understanding the regularity of the Martin kernel, which is a delicate problem in the manifold setting. This will be treated in a subsequent paper ([LS3]).

Note that the Hausdorff dimension of the distribution of r m λ x , denoted by dim H r m λ x , is given by h λ {pκ λ q for a fixed number κ associated with the distance function on the boundary (see (3.1)) ( [L1]). The following is a corollary of Theorem 1.1 and Theorem 1.6.

Corollary 1.7. Let M be a closed connected smooth manifold. For any C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q and all x P Ă M , the function λ Þ Ñ dim H r m λ x is C 1 differentiable.

If we switch from a negatively curved manifold to a finitely generated hyperbolic group G, we do not have to control any more the subtle influence of the changes of soft geometric structures. But diffusions live on in the form of random walks, so the regularity problem of random dynamics with respect to probabilities still has its interest. More precisely, let N be the set of probability measures with support a fixed finite subset G 0 Ă G which generates G as a semigroup. Then N is an open finite dimensional simplex, in particular, it has a natural real analytic structure. Each element µ P N defines a random walk on G by convolutions tµ pnq u nPN . The linear drift and the entropy of µ are defined by µ :" lim where, for γ P G, |γ| denotes the word length of γ. In this setting, much progress has been achieved in understanding the regularity of µ , h µ with respect to µ: the continuity property was considered by Erschler and Kaimanovich ([EK]), the Lipschitz property was shown by one of the authors ( [L5]), the differentiability under one parameter family of differentiable curve of µ is due to Mathieu ([Mat]), and, more recently, the real analytic property is shown by Gouëzel ([Go]). (See [Go] for the whole history and other previous results in various settings.) In the same flavor of the rigidity problems in the manifold case, a basic question is what can we say about the group structure using our knowledge of the dynamical quantities µ and h µ ? We don't have an answer to this general question, but we can mention one result which is related to pbq of (1.1) in the above group setting: in [GMM], Gouëzel, Mathéus and Maucourant show that if G is not virtually free, then there is c ă 1 such that for any symmetric measure µ P N , h µ ď c µ υ, where υ denotes the volume entropy of the group in the word metric.

We arrange the paper as follows. In Section 2, we give some preliminaries. In Section 3, we assume Theorem 1.3 and prove consecutively Theorem 1.2, Theorem 1.1, Theorem 1.4 and Theorem 1.5. Section 4 is for the Eells-Elworthy-Malliavin construction of the stochastic flow corresponding to the Brownian motion and its related dynamical properties. The estimations of the growth of various stochastic tangent structures are done with some special care since we are in the non-compact case. The strategy for proving the first order differentiability in Theorem 1.3 and the i " 1 case of (1.4) and (1.5) is explained in Section 5.1. Section 5 is devoted to the details of that proof: Section 5.2 is for the construction for the C 1 regularity of z λ,1 T , followed by the existence proof and estimations in Section 5.3-5.5, and the proof of Theorem 1.3 with i " 1 is given in Section 5.6 using the regularities and estimations of z λ,1 T . The rest of the proof of Theorem 1.3 is by induction on the order of differentiability. See Section 6.1 for the description of the necessary steps and Section 6.2 for their proofs. Finally, in Section 7, we consider the first order regularity of the entropy.

Preliminaries

In this section, we introduce the basic notions related to formula (1.2). In the rest of the paper, if it is not specified, we only consider the elements of M k pM q, k pM q with k ě 3.

2.1. Jacobi fields and the geodesic flow. For g P M k pM q, let ∇, R be the Levi-Civita connection and the curvature tensor on pM, gq and p Ă M , r gq. Recall that a unit speed r ggeodesic t Þ Ñ γptq P Ă M is such that ∇ 9 γ 9 γ " 0, where 9 γptq " ∇ B Bt γptq. The Jacobi fields along γ are vector fields t Þ Ñ Jptq P T γptq Ă M which describe the infinitesimal variations of the geodesics around γ. It is well known that Jptq satisfies the Jacobi equation ∇ 9

γptq ∇ 9 γptq Jptq `RpJptq, 9 γptqq 9 γptq " 0 and is uniquely determined by the values of Jp0q and J 1 p0q. Let N pγq be the normal bundle of γ, i.e., N pγq :" ď tPR N t pγq, where N t pγq " Y P T γptq Ă M : xY, 9 γptqy " 0 ( .

A p1, 1q-tensor along γ is a family V " tV ptq, t P Ru, where each V ptq is an endomorphism of N t pγq such that for any family Y t of parallel vectors along γ, the covariant derivative ∇ 9 γptq pV ptqY t q exists. The curvature tensor R induces a symmetric p1, 1q-tensor along γ by RptqY " RpY, 9 γptqq 9 γptq. A p1, 1q-tensor V ptq along γ is called a Jacobi tensor if it satisfies ∇ 9 γptq ∇ 9 γptq V ptq `RptqV ptq " 0.

If V ptq is a Jacobi tensor along γ, then V ptqY t is a Jacobi field for any parallel field Y t along γ.

The Jacobi fields can also be visualized using the geodesic flow map on the unit tangent bundle. For x P Ă M and v P T x Ă M , an element w P T v T Ă M is vertical if its projection on T x Ă M vanishes. The vertical subspace V v is identified with T x Ă M . The connection defines a horizontal complement H v , which also can be identified with T x Ă M . This gives a horizontal/vertical Whitney sum decomposition

T T Ă M " T Ă M ' T Ă M .
Define the inner product on T T Ă M by @ pY 1 , Z 1 q, pY 2 , Z 2 q D r g :"

@ Y 1 , Y 2 D r g `@Z 1 , Z 2 D r g .
It induces a Riemannian metric on T Ă M , the so-called Sasaki metric. The unit tangent bundle S Ă M of the universal cover p Ă M , r gq is a subspace of T Ă M with tangent space T px,vq S Ă M " pY, Zq : Y, Z P T x Ă M , Z K v ( , for x P Ă M , v P S x Ă M .

Assume v " px, vq P S Ă M and let γ v be the r g-geodesic starting at x with initial velocity v. Horizontal vectors in T v S Ă M correspond to pairs pJp0q, 0q. In particular, the geodesic spray X v at v is the horizontal vector associated with pv, 0q. A vertical vector in T v S Ă M is a vector tangent to S x Ă M , the set of unit tangent vectors at x. It corresponds to a pair p0, J 1 p0qq, with J 1 p0q orthogonal to v. The orthogonal space to X v in T v S Ă M corresponds to pairs pv 1 , v 2 q, v i P N 0 pγ v q for i " 1, 2.

The vector field tX v u vPS Ă M generates the geodesic flow tΦ t u tPR on the unit tangent bundle, where Φ t : S Ă M Ñ S Ă M , v Þ Ñ 9 γ v ptq. Any Jacobi field along a geodesic γ v is of the form DΦ t pwq, where w P T v S Ă M is an infinitesimal change of the initial point v. More explicitly, if pJp0q, J 1 p0qq is the horizontal/vertical decomposition of w P T v S Ă M , then pJptq, J 1 ptqq is the horizontal/vertical decomposition of DΦ t pwq P T Φtpvq S Ă M .

2.2. Anosov flow and invariant manifolds. Assume g P k pM q. The r g-geodesic flow Φ t on S Ă M has some special properties due the negative curvature nature of the space.

Firstly, p Ă M , r gq has no conjugate points. Hence we can identify S Ă M with Ă M ˆB Ă M since each pair px, ξq P Ă M ˆB Ă M corresponds to a unique unit speed geodesic γ x,ξ , which begins at x and is asymptotic to ξ, and the mapping B Ă M Þ Ñ S x Ă M sending ξ to 9 γ x,ξ p0q is a bijection. In the p Ă M , B Ă M q-coordinate, the geodesic flow map Φ t has the expression Φ t px, ξq " pγ x,ξ ptq, ξq, @px, ξq P S Ă M .

Furthermore, the geodesic flow on S Ă M is Anosov : the tangent bundle T S Ă M decomposes into the Witney sum of three DΦ t -invariant subbundles E c ' E ss ' E su , where E c is the 1-dimensional subbundle tangent to the flow and E ss and E su are the strongly contracting and expanding subbundles, respectively, so that there are constants C, c ą 0 such that i) }DΦ t w} ď Ce ´ct }w} for w P E ss , t ą 0. ii) }DΦ ´1 t w} ď Ce ´ct }w} for w P E su , t ą 0.

The E ss , E su and E c are the so-called stable, unstable and central bundles, respectively.

The subbundles E ss , E su have their characterizations using Jacobi tensors. Assume v " px, vq P S Ă M . For each s ą 0, let S v,s be the Jacobi tensor along γ v with the boundary conditions S v,s p0q " Id and S v,s psq " 0. Since p Ă M , r gq has no conjugate points, the limit lim sÑ`8 S v,s ": S v exists ( [Esc]) and is called the stable tensor along the geodesic γ v . Similarly, by reversing the time s, we obtain the unstable tensor U v along the geodesic γ v . The stable subbundle E ss at v is the graph of the mapping S 1 v p0q, considered as a map from

N 0 pγ v q to V v sending Y to S 1 v p0qY
, where N 0 pγ v q :" tw, w P H v , w K X v u. Similarly, the unstable subbundle E su at v is the graph of the mapping U 1 v p0q considered as a map from N 0 pγ v q to V v .

Due to the Anosov property of the geodesic flow, the distributions of E ss , E su (and hence E c ' E ss , E c ' E su ) are Hölder continuous ( [START_REF] Anosov | Tangent fields of transversal foliations in U -systems[END_REF], see also [START_REF] Ballmann | Lectures on spaces of nonpositive curvature, With an appendix by Misha Brin[END_REF]Proposition 4.4]). Hence, the p1, 1q-tensors S v , S 1 v , U v and U 1 v are also Hölder continuous with respect to v. Associated with the bundle E cs :" E c ' E ss are the (weak) stable manifolds of Φ t :

(2.1) The geodesic flow Φ t on S Ă M naturally descends to the geodesic flow Φ t on g-unit tangent bundle SM , carrying the tangent splitting and the corresponding submanifolds downstairs. Indeed, the action of G on the tangent bundle E (where E denotes any one of E ss , E su and E c ) satisfies ψpEpx, ξqq " EpDψpx, ξqq for all ψ P G so that it defines the DΦ t -invariant subbundles E ss , E su and E c of T SM , the so-called stable, unstable and central bundles. We see that E c is tangent to the flow direction and E ss , E su are such that i) }DΦ t w} ď Ce ´ct }w} for w P E ss , t ą 0. ii) }DΦ ´1 t w} ď Ce ´ct }w} for w P E su , t ą 0.

Similarly, the action of G on the submanifolds W (where W denotes any one of W s , W ss , W u and W su ) satisfies ψpW px, ξqq " W pDψpx, ξqq for all ψ P G so that it defines the stable, strong stable, unstable and strong unstable manifolds of the geodesic flow on SM , which have tangents E ss ' E c , E ss , E su ' E c and E su , respectively. In particular, the collection of W s px, ξq defines a foliation W " tW s pvqu vPSM on SM , the so-called stable foliation of SM . Each W s px, ξq can be identified with Ă M ˆtξu. Hence the quotients W s pvq are naturally endowed with the Riemannian metric induced from r g. They are C k´1 immersed submanifolds of SM depending continuously on v in the C k´1 topology ( [SFL]).

2.3. Harmonic measure for the stable foliation. We continue to assume g P k pM q. Associated with the stable foliation W is the harmonic measure which is closely related to the leafwise Brownian motion. Write ∆ W for the leafwise Laplace operator of W, which acts on functions that are of class C 2 along the leaves of W. A probability measure m on SM is called harmonic if it satisfies, for any

C 2 function f on SM , ż SM ∆ W f dm " 0.
Since pM, gq is negatively curved, there is a unique harmonic measure m associated to the stable foliation ( [Ga]). Let r m be the G-invariant extension of m to S Ă M . It is closely related to the Brownian motion on the stable leaves. For px, ξq P S Ă M , let ppt, px, ξq, dpy, ηqq :" ppt, x, yq dVol r g pyqδ ξ pηq, where δ ξ pηq is the Dirac function at ξ. Then p is just the transition probability function of the Brownian motion on W s px, ξq " Ă M ˆtξu starting from px, ξq. Let r Ω `be the space of continuous paths ω : r0, `8q Ñ S Ă M equipped with the smallest σ-algebra for which the projections R t : ω Þ Ñ ωptq are measurable. Let tP px,ξq u be the corresponding Markovian family of p on Ω `. Then for every t ą 0 and every Borel set A Ă Ă M ˆB Ă M , r P px,ξq ptω P Ω `: ωptq P Auq " ż A ppt, px, ξq, dpy, ηqq.

Proposition 2.1. ( [Ga]) The following hold true.

i) The measure r m satisfies, for any

f P C 2 p Ă M ˆB Ă M q with compact support, ż Ă M ˆB Ă M ˆż Ă M ˆB Ă M f py, ηqppt, px, ξq, dpy, ηqq ˙d r mpx, ξq " ż Ă M ˆB Ă M f px, ξq d r mpx, ξq.
ii) The measure r P " ş r P px,ξq d r mpx, ξq on r Ω `is invariant under every t-time shift mapping σ t : r Ω `Ñ r Ω `, σ t pr ωpsqq " r ωps `tq, for s ą 0 and r ω P r Ω `. iii) The measure r m can be expressed locally at px, ξq P Ă M ˆB Ă M as d r m " dx ˆd r m x , where dx is proportional to the volume element and r m x is the hitting probability at B Ă M of the Brownian motion starting at x.

The group G acts naturally and discretely on the space r Ω `with quotient the space Ω òf continuous paths in SM , and this action commutes with the shift σ t , t ě 0. Therefore, the measure r P is the extension of a finite, shift invariant measure P on Ω `. We identify SM with M 0 ˆB Ă M , where M 0 is a connected fundamental domain of p Ă M , r gq. Hence we can also identify Ω `with the lift of its elements in r Ω `starting from M 0 . We will continue to denote elements in Ω `by ω and will clarify the notation whenever there is an ambiguity.

In this paper, we normalize the harmonic measure m to be a probability measure, so that P is also a probability measure. We denote by E P the corresponding expectation symbol.

A nice property for the laminated Brownian motion is that the semi-group σ t , t ě 0, of transformations of Ω `has strong ergodic properties with respect to the probability P.

Proposition 2.2. ( [Ga], cf. [START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF]Proposition 2.3]) The shift semi-flow σ t , t ě 0, is mixing on pΩ `, Pq in the sense for any bounded measurable functions f 1 , f 2 on Ω `, lim tÑ`8 E P pf 1 pf 2 ˝σt qq " E P pf 1 qE P pf 2 q.

2.4. Busemann function and the linear drift. In this subsection, we derive (1.2).

Let g P k pM q. For v " px, ξq P M 0 ˆB Ă M , the projection on Ă M of the law of P v on W s px, ξq " Ă M ˆtξu is the same as that of P x of the Brownian motion on Ă M starting from x. For ω P Ω `, we still denote by ω its projection to Ă M . By ergodicity of P with respect to the shift map σ t (Proposition 2.2), for P-almost all path ω P Ω `, its leafwise linear drift coincides with .

Since r

g is negatively curved, for P-almost all path ω, ωptq tends to a point in the geometric boundary B Ă M ([Kai1]). Write ωp8q :" lim tÑ`8 ωptq. Roughly speaking, ω follows γ ωp0q,ωp8q . Hence the drift of ωptq from ωp0q can be measured via its shadow on γ ωp0q,ωp8q . A candidate function for this measurement is the Busemann function. Let x 0 P Ă M be a reference point. For y, z P Ă M , define b x 0 ,y pzq :" dpz, yq ´dpx 0 , yq.

The assignment of y Þ Ñ b x 0 ,y is continuous, one-to-one and takes value in a relatively compact set of functions for the topology of uniform convergence on compact subsets of Ă M . The Busemann compactification of Ă M is the closure of Ă M for that topology ( [BGS]) and it coincides with the geometric compactification in the negative curvature case (see [Ba]). So for each v " px, ξq P Ă M ˆB Ă M , the function

b v pzq :" lim yÑξ b x,y pzq, for z P Ă M ,
is well-defined and is called the Busemann function at v. It is known ( [EO]) that, if we consider b v as a function defined on W s px, ξq, then

(2.3) ∇b v pzq " ´Xpz, ξq.
The difference between b v pyq and b v py 1 q is preserved when py, ξq and py 1 , ξq are driven by the geodesic flow Φ t . Hence

W ss pvq " tpy, ξq : b v pyq " b v pxqu . Note that W ss pvq locally is a C k´1 graph from E ss v to E c v ' E su v and is tangent to E ss v .
So, by the Jacobi tensor characterization of E ss v and (2.3), it is true ( [Esc, HIH]) that ∇ w p∇b v qpxq " ´S1 v p0qpwq, @w P T x Ă M .

Thus,

(2.4) ∆ x b v " ´DivX " ´Trace of S 1 v p0q, which is the mean curvature of the set of footpoints of W ss px, ξq. Note that for each ψ P G, b px 0 ,ψξq pψxq " b px 0 ,ξq pxq `bpψ ´1x 0 ,ξq px 0 q.

Hence ∆ x b px 0 ,ξq satisfies ∆ ψx b px 0 ,ψξq " ∆ x b px 0 ,ξq and defines a function B on the unit tangent bundle SM , which is called the Laplacian of the Busemann function. The function B is a Hölder continuous function on SM by the Hölder continuity of the strong stable tangent bundles ( [START_REF] Anosov | Tangent fields of transversal foliations in U -systems[END_REF], see Section 2.2). Now, we can derive the integral formula of the linear drift using the geodesic spray and the harmonic measure [START_REF] Kaimanovich | Brownian motion and harmonic functions on covering manifolds. An entropic approach[END_REF]). For P-almost all path ω P Ω `, let v :" ωp0q and η :" ωp8q P B Ă M . When t goes to infinity, the process b v pωptqq ´dpx, ωptqq converges P-a.e. to the a.e. finite number ´2pξ|ηq x , where the Gromov product p¨|¨q x is such that (2.5) pξ|ηq x :" lim yÑξ,zÑη py|zq x and py|zq x :" 1 2 pdpx, yq `dpx, zq ´dpy, zqq .

So for P-almost all ω P Ω `, we have

lim tÑ`8 1 t b v pωptqq " .
Using the fact that the leafwise Brownian motion has generator ∆ and is ergodic with invariant measure m on SM , we obtain

" lim tÑ`8 1 t ż t 0 B Bs b v pωpsqq ds " lim tÑ`8 1 t ż t 0 ∆b v pωpsqq ds ˆ" ż M 0 ˆB Ă M ∆b v dm " ´żM 0 ˆB Ă M Div W pXq dm, (2.6)
where Div W is the laminated divergence operator for the stable foliation W. Since on each leaf we have Div W pXq " DivpXq, (2.6) reduces to (1.2). But that will not simplify the discussion of the regularity of the linear drift under metric changes since DivpXq is essentially a leafwise object. In contrast, (2.6) is more suitable for this purpose because of the natural connection between the geodesic spray X and the geodesic flow.

Regularity of the linear drift

In this section, we assume Theorem 1.3 holds true. We first prove Theorem 1.1 by showing the regularities of Div W X and m under a one parameter family of C k deformation of metrics in k pM q and then prove Theorems 1.4 and 1.5.

3.1. Regularity of the leafwise divergence term Div W X. Clearly, the geodesic sprays of a metric g P k pM q form a C k´1 vector field which varies C k´1 with respect to C k metric change. But this does not imply the regularity of Div W X with respect to the metric changes since we are considering the leafwise divergence.

Laminate S Ă

M " Ă M ˆB Ă M into stable leaves tW s px, ξq " Ă M ˆtξuu, where each leaf can be identified with p Ă M , r gq, but is only Hölder continuous in the ξ-coordinate (see Section 2.2). Consequently, Xpy, ξq P T W s px, ξq is C k´1 in the y-coordinate, but is only Hölder continuous in the ξ-coordinate. Let g 1 P k pM q be another metric. Its geometric boundary B Ă M g 1 can be identified with B Ă M . But the r g 1 -geodesic spray X r g 1 px, ξq differs from Xpx, ξq and the divergence operator on the r g 1 -stable leaf W s r g 1 px, ξq differs from that on W s px, ξq. Both difference contribute to the change of pDiv W Xqpx, ξq in metrics. This, by (2.4), can be understood by a study of the regularity of X and E ss in k pM q.

Assume g P k pM q. The set of r g-oriented geodesics in Ă M can be identified with

B 2 Ă M :" pB Ă M ˆB Ă M qztpξ, ξq : ξ P B Ă M u.
Indeed, for px, ξq P S Ă M , let γ : R Þ Ñ Ă M be the unique geodesic with 9 γp0q " px, ξq and write B `γ :" lim tÑ`8 γptq and B ´γ :" lim tÑ´8 γptq. The mapping γ Þ Ñ pB `γ, B ´γq establishes a homeomorphism between the set of all oriented geodesics in p Ă M , r gq and B 2 Ă M . Consequently, for any g 1 P k pM q, the mapping D g 1 :

B 2 p Ă M q Ñ B 2 p Ă M r
g 1 q induced from the identity isomorphism from G to itself can be viewed as a homeomorphism between the set of oriented geodesics in p Ă M , r gq and p Ă M , r g 1 q. Further realize points from S Ă M r g 1 by pairs pγ, yq, where γ is an oriented geodesic and y P γ. For g 1 close to g, we obtain a map r

F g 1 : S Ă M Ñ S Ă M r g 1 which sends pγ, yq P S Ă M to r F g 1 pγ, yq " pD g 1 pγq, y 1 q,
where y 1 is the unique intersection point of D g 1 pγq and the hypersurface texp r g Y : Y K vu with v being the vector in S y Ă M pointing at B `γ. The map r F g 1 is a homeomorphism between S Ă M and S Ă M r g 1 which preserves the geodesics, i.e., sending r g-geodesics to r g 1 -geodesics, and is referred to as a pr g, r g 1 q-Morse correspondence map. The restriction of r

F g 1 to geodesics asymptotic to ξ P B Ă M is a homeomorphism from W s r g px, ξq to W s r g 1 px, ξq. Let r π g 1 : S Ă M r g 1 Ñ S Ă M be the map sending v to v{}v} r g which records the direction information points of S Ă M r g 1 in S Ă M . Then r π g 1 ˝r F g 1 is a homeomorphism between S Ă M and itself.
The map r F g 1 induces a homeomorphism F g 1 between SM and SM g 1 which sends ggeodesics to g 1 -geodesics and is called a pg, g 1 q-Morse correspondence map. For any sufficiently small , if g 1 is sufficiently close to g, then F g 1 is such that the footpoint of F g 1 pvq belongs to the hypersurface of points texp g Y : Y K v, }Y } g ă u, where v is the projection of v in SM . Let π g 1 : SM g 1 Ñ SM be the natural projection map sending v to v{}v} g . Then π g 1 ˝Fg 1 is a homeomorphism between SM and itself.

For g 1 in a small neighborhood of g in k pM q, let E g 1 (resp. E g 1 ) denote any one of E ss g 1 , E su g 1 and E c g 1 (resp. any one of E ss g 1 , E su g 1 and E c g 1 ). We also regard E g 1 (resp. E g 1 ) as a mapping from S Ă M r g 1 (resp. SM g 1 ) to its tangent bundle. Of our special interest, is the regularity of the mappings

g 1 Þ Ñ r π g 1 ˝r F g 1 , g 1 Þ Ñ Dr π g 1 ˝Eg 1
. Equivalently, we can consider the regularity of the downstairs mappings g 1 Þ Ñ π g 1 ˝Fg 1 and g 1 Þ Ñ Dπ g 1 ˝Eg 1 , for which, we can take advantage of the compactness of M to construct certain manifolds of maps so that the implicit function theory applies ( [LMM, KKPW]).

Let H k´1 pSM q be the Banach space of C k´1 vector fields on SM endowed with the topology of uniform C k´1 convergence on compact subsets. Let X g be the vector field generating the g-geodesic flow. Then X g 1 , the projection (via Dπ g 1 ) of the generating vector field of the g 1 -geodesic flow on SM g 1 , belongs to H k´1 pSM q and is C k´1 close to X g whenever g 1 is C k close to g. For α P r0, 1q, let C α pSM, N q denote the Banach space of α-Hölder (or continuous for α " 0) maps from SM to a Banach space N endowed with the topology given by the α-Hölder norm on SM . Consider

C α Φ pSM, SM q :" " F P C α pSM, SM q : D Φ F pvq :" d dt F pΦ t pvqq ˇˇˇt "0
exists and is α-Hölder * with the topology of the norm }F } `}D Φ F } α , where } ¨}α denotes the α-Hölder norm, together with the mapping Ψ :

H k´1 pSM q ˆCα Φ pSM, SM q ˆCα pSM, Rq Ñ C α pSM, T SM q ΨpY, F, f q " Y ˝F ´f ¨DΦ F.
By hyperbolicity of the g-geodesic flow Φ t , the implicit function theory applies to Ψ if we further require F P C α Φ pSM, SM q to be such that the footpoint of F pvq lies in texp g pwq : w K vu for any v P SM . The following structural stability theorem is due to de la Llave-Marco-Moriyón ( [LMM]) for continuous case and Katok-Knieper-Pollicott-Weiss ( [KKPW]) for Hölder continuous case.

Proposition 3.1. ([KKPW, Proposition 2.2]) For g P k pM q, there exist α P p0, 1q and a neighborhood U Ă H k´1 pSM q of X g and

C k´2 maps U Ñ C α Φ pSM, SM q : Y Þ Ñ F Y and U Ñ C α `SM, r 1 2 , `8q ˘: Y Þ Ñ f Y such that Y ˝FY " f Y D Φ F . Moreover, the maps U Ñ C 0 Φ pSM, SM q : Y Þ Ñ F Y and U Ñ C 0 `SM, r 1 2 , `8q ˘: Y Þ Ñ f Y are C k´1 . Define C α pS Ă M , N q, C α Φ pS Ă M , N q analogously as C α pSM, N q, C α Φ pSM, N q.
A consequence of Proposition 3.1 is Corollary 3.2. Assume g P k pM q. There exist α P p0, 1q and a neighborhood V of g in k pM q such that the map

g 1 P V Þ Ñ π g 1 ˝Fg 1 is C k´2 into C α Φ pSM, SM q and is C k´1 into C 0 Φ pSM, SM q; the map g 1 P V Þ Ñ r π g 1 ˝r F g 1 is C k´2 into C α Φ pS Ă M , S Ă M q and is C k´1 into C 0 Φ pS Ă M , S Ă M q.
The regularity of g 1 Þ Ñ Dπ g 1 ˝Eg 1 and g 1 Þ Ñ Dr π g 1 ˝Eg 1 can be analyzed analogously ( [Con]). Let G be the Grassmann bundle of u-planes on T SM , where u " dimE su g . Let C α Φ pSM, Gq be the space of α-Hölder maps p F : SM Ñ G, p F pvq " pF pvq, Epvqq, where F P C α Φ pSM, SM q, with the topology of the α-Hölder norm on F, D Φ F and E. Then instead of Ψ, one can consider the maps

p Ψ ˘: H k´1 pSM q ˆCα Φ pSM, Gq ˆCα pSM, Rq Ñ C α pSM, T SM ' Gq p Ψ ˘pY, p F , f q " `Y ˝F ´f ¨DΦ F, Dψ τ Y pvq ˝F pΦ ˘1pvqqEpΦ ˘1pvqq ˘,
where ψ t is the time t map of the flow generated by Y and τ Y is the time change such that

ψ τ Y pvq ˝FY pΦ ˘1pvqq " F Y pvq, @v P SM.
Again, by hyperbolicity of the flow generated by Y which is close to X g and the invariance of the corresponding strong stable and unstable bundles, denoted by E ss Y , E su Y , the implicit function theory applies for p Ψ `, p Ψ ´and gives the following.

Proposition 3.3. ([Con, Proposition 2.1]) For g P k pM q, there exist a neighborhood U of X g in H k´1 pSM q and α P p0, 1q such that the map

U Ñ C α Φ pSM, Gq : Y Þ Ñ pv Þ Ñ E Y ˝FY pvqq is C k´3 and the map U Ñ C 0 Φ pSM, Gq : Y Þ Ñ E Y ˝FY is C k´2 , where E Y " E ss Y or E su Y .
Let r G be the Grassmann bundle of u-planes on T S Ă M (where u " dimE su g ) and define

C α Φ pS Ă M , r Gq in analogy with C α Φ pSM, Gq.
The following is an application of Proposition 3.3 to the geodesic flows.

Corollary 3.4. There exist α P p0, 1q and a neighborhood V of g in k pM q such that the map

g 1 P V Þ Ñ Dπ g 1 ˝Eg 1 ˝Fg 1 is C k´3 into C α Φ pSM, Gq and is C k´2 into C 0 Φ pSM, Gq
, where E g 1 is any one of E ss g 1 , E su g 1 and E c g 1 . Similarly, the map

g 1 P V Þ Ñ Dr π g 1 ˝Eg 1 ˝r F g 1 is C k´3 into C α Φ pS Ă M , r Gq and is C k´2 into C 0 Φ pS Ă M , r
Gq, where E g 1 is any one of E ss g 1 , E su g 1 and E c g 1 .

For λ P p´1, 1q Þ Ñ g λ P k pM q, we write X λ for the r g λ -geodesic spray, pE ss q λ for the r g λ -stable bundle and Div λ for the divergence operator associated with the r g λ -stable foliation.

Proposition 3.5. Let g P k pM q. There exist α P p0, 1q and a neighborhood V g of g in k pM q such that for any

C k curve λ P p´1, 1q Þ Ñ g λ P V g with g 0 " g, i) λ Þ Ñ X λ is C k´3 into C α p Ă M ˆB Ă M , T T Ă M q and is C k´2 into C 0 p Ă M ˆB Ă M , T T Ă M q, ii) λ Þ Ñ pE ss q λ is C k´3 into C α p Ă M ˆB Ă M , r Gq and is C k´2 into C 0 p Ă M ˆB Ă M , r Gq, and iii) λ Þ Ñ Div λ X λ is C k´3 into C α p Ă M ˆB Ă M , R `q and is C k´2 into C 0 p Ă M ˆB Ă M , R `q.
Proof. Express the pr g, r g λ q-Morse correspondence map r

F g λ from Ă M ˆB Ă M to itself as r F λ px, ξq " `f λ ξ pxq, ξ ˘, @px, ξq P Ă M ˆB Ă M ,
where f λ ξ records the change of the footpoint for the unit vector pointing at ξ in the boundary. For px, ξq P Ă M ˆB Ă M , we transform X r g px, ξq to X r g λ px, ξq in three steps: the first is to follow the footpoint of the inverse of the pg λ , gq-Morse correspondence from X r g px, ξq to X r g ppf λ ξ q ´1pxq, ξq with the constraint that the vector remains within T W s px, ξq; the second is to use the pg λ , gq-Morse correspondence from X r g ppf λ ξ q ´1pxq, ξq to X r g λ px, ξq{}X r g λ px, ξq} r g ; the third is to adjust the length of X r g λ px, ξq{}X r g λ px, ξq} r g to be 1 in the metric r g λ . Hence, X r g λ px, ξq ´Xr g px, ξq

" ˜Xr g λ px, ξq ´Xr g λ px, ξq }X

r g λ px, ξq} r g ¸`˜X r g λ px, ξq }X r g λ px, ξq} r g ´Xr g ppf λ ξ q ´1pxq, ξq
Xr g ppf λ ξ q ´1pxq, ξq ´Xr g px, ξq

":

paq λ `pbq λ `pcq λ .
Note that paq 0 , pbq 0 and pcq 0 are all zero. So the regularity of λ Þ Ñ X λ will follow from that of paq λ , pbq λ and pcq λ by Taylor's formula. This is true since paq λ corresponds to length change and is C k in λ, pbq λ is C k´2 (or C k´3 ) in λ depending on α " 0 (or not) by Corollary 3.2, while pcq λ has the same regularity as pbq λ since Xpx, ξq is C k´1 in the x-coordinate.

Similarly, we write v λ " X λ px, ξq and pE ss q λ pv λ q ´pE ss q 0 pv 0 q " ´pE ss q λ pv λ q ´pE ss q 0 ppf λ ξ q ´1pxq, ξq pE ss q 0 ppf λ ξ q ´1pxq, ξq ´pE ss q 0 pv 0 q ":

pdq λ `peq λ .
This means we can transport pE ss q 0 pv 0 q to pE ss q λ pv λ q in two steps: first is to transport pE ss q 0 pv 0 q to pE ss q 0 ppf λ ξ q ´1pxq, ξq along the tangent bundle of W s px, ξq and follow the footpoint of the inverse of the pr g λ , r gq-Morse correspondence; the second is to use the Morse correspondence for the stable bundle from pE ss q 0 ppf λ ξ q ´1pxq, ξq to pE ss q λ pv λ q. Note that pdq 0 , peq 0 are zero. The regularity of λ Þ Ñ pE ss q λ will follow from that of pdq λ , peq λ by Taylor's formula, which will follow by Corollary 3.4 if we can show the C k´1 dependence of E ss px, ξq on the x-coordinate. This is true because each E ss py, ξq is the tangent plane of the strong stable manifold W ss py, ξq. Locally, W ss px, ξq is a C k´1 graph from E ss px,ξq to E c px,ξq ' E su px,ξq . This means, locally, y Þ Ñ E ss py, ξq is C k´1 along the leaf W ss px, ξq. On the other hand, by invariance of the strong stable bundle with respect to the geodesic flow, y Þ Ñ E ss py, ξq is smooth as y varies on the geodesic passing through x asymptotic to ξ. By invariance of the strong stable leaf under the geodesic flow, W ss px, ξq and the time direction (i.e. the direction of the geodesic spray) consist of a coordinate chart for W s px, ξq. This shows, locally at x, y Þ Ñ E ss py, ξq is C k´1 along W s px, ξq " Ă M ˆtξu.

Finally iii) is just an application of ii) noting that for any g P k pM q, we have

pDivXqpx, ξq " Trace of S 1 v p0q, @v " px, ξq P Ă M ˆB Ă M ,
and the stable bundle E ss at v is the graph of the mapping S 1 v p0q, considered as a map from N 0 pγ v q to V v sending Y to S 1 v p0qY , where N 0 pγ v q :" tw, w P H v , w K X v u.

3.2. Regularity of the harmonic measure. In this subsection, we prove Theorem 1.2 following the sketch that we gave in the Section 1.

For g λ P k pM q, we introduce a metric on B Ă M as follows. Let κ ą 0. For x P Ă M , define

(3.1) d κ,λ x pζ, ηq :" e ´κpζ|ηq λ x , @ζ, η P B Ă M ,
where p¨|¨q λ x is the Gromov product defined in (2.5) for d

r g λ . If κ is small, each d κ,λ x p¨, ¨q defines a distance on B Ă M , the so-called κ-Busemann distance ([Kai2]), which is related to the r g λ -Busemann functions b λ since (3.2) b λ v pyq " lim ζ,ηÑξ ´pζ|ηq λ y ´pζ|ηq λ x ¯, for any v " px, ξq P S Ă M , y P Ă M . Let b ą 0. For continuous functions f on SM " M 0 ˆB Ă M , define }f } λ b :" sup x,ξ | r f px, ξq| `sup x,ξ 1 ,ξ 2 | r f px, ξ 1 q ´r f px, ξ 2 q|e bpξ 1 |ξ 2 q λ x .
Let H λ b be the Banach space of continuous functions f on SM with }f } λ b ă `8. Elements of H λ b are continuous on SM and Hölder continuous with respect to the direction changes. Recall that the transition probability of the r g λ -Brownian motion on the stable leaf W s r g λ px, ξq " Ă M ˆtξu starting from px, ξq is given by p λ pt, px, ξq, dpy, ηqq :" p λ pt, x, yq dVol λ pyqδ ξ pηq, where tp λ pt, x, ¨qu xP Ă

M ,tPR

`is the transition probabilities of the r g λ -Brownian motion on Ă M , δ ξ pηq is the Dirac function at ξ and Vol λ is the r g λ volume element. Then p λ descends to be the transition probability of g λ -Brownian motion the stable leaves of SM : for px, ξq, py, ηq P SM " M 0 ˆB Ă M , the transition probability is q λ pt, px, ξq, dpy, ηqq " ÿ βPG p λ pt, px, ξq, dpβy, βηqq " ÿ βPG p λ pt, x, βyqdVol λ pyqδ ξ pβηq.

Let Q λ t pt ě 0q be given in (1.3). It defines the action of r0, `8q on continuous functions f on SM which describes the ∆ W g λ -diffusion. It was shown in [L3] that for sufficiently small b ą 0, there exists T ą 0 such that Q λ T is a contraction on H λ b and hence, as t Ñ 8, Q λ t converges to the mapping f Þ Ñ ş f dm λ exponentially in t for f P H λ b . Thus, each harmonic measure m λ is a fixed point of the dual operation pQ λ T q ˚in the dual space pH λ b q ˚with the weak topology, where pQ λ T q ˚pµqpf q :" µpQ λ T pf qq, for all µ P pH λ b q ˚, f P H λ b .

The following proposition shows that H λ b can be chosen to be independent of g λ . Proposition 3.6. Let V g be as in Theorem 1.3. For every b ą 0 small enough, there exist C ą 0 and k ă 1 such that, for all λ P p´1, 1q, t ą 0 and f

P H 0 b , › › › › Q λ t f ´ż f dm λ › › › › b ď Ck t }f } b .
The proof of Proposition 3.6 follows [START_REF] Ledrappier | Central Limit Theorem in negative curvature[END_REF]Theorem 3] for an individual metric. The only modification is to find a common Hölder continuous function space independent of the metrics where the contractions (of Hölder norm) happen. Denote d and pξ ˇˇηq x for the r g 0 distance and its Gromov product. The key lemma is the following.

Lemma 3.7. Let V g be as in Theorem 1.3. There is a number b 1 ą 0 such that for any b, 0 ă b ă b 1 , there exists k 1 ă 1 such that for t large enough, x P M 0 and all ξ, η, ξ " η, we have for all λ P p´1, 1q,

E λ x,ξ ˆe´b ´pξ|ηq tx t s λ ´pξ|ηqx ¯˙ă k t 1 ,
where tx t s λ denotes the r g λ -Brownian motion on W s px, ξq starting from px, ξq and E λ

x,ξ denotes its corresponding expectation.

As a preparation for the proof of Lemma 3.7, define on M 0 ˆB Ă M ˆB Ă M the transition probabilities q 2,λ pt, px, ξ 1 , ξ 2 q, dpy, η 1 , η 2 qq :" ÿ βPG p λ pt, x, βyq dVol λ pyqδ ξ 1 pβη 1 qδ ξ 2 pβη 2 q and the corresponding operator Q 2,λ t on continuous functions on

M 0 ˆB Ă M ˆB Ă M : Q 2,λ t f px, ξ 1 , ξ 2 q " ż f py, η 1 , η 2 qq 2,λ ppx, ξ 1 , ξ 2 q, dpy, η 1 , η 2 qq .
By analogy with the case of Q λ t , there is a unique Q 2,λ t -invariant probability measure on M 0 ˆB Ă M ˆB Ă M which is related to the harmonic measure m λ as follows.

Lemma 3.8. ([L3, Proposition 1]) For each g λ P k pM q, with the above notations, there is a unique probability measure m 2,λ on

M 0 ˆB Ă M ˆB Ă M satisfying ż Q 2,λ t f dm 2,λ " ż f dm 2,λ
for all f P CpM 0 ˆB Ă M ˆB Ă M , Rq and all positive t. The measure m 2,λ is characterized by

ż f dm 2,λ " ż M 0 ˆB Ă M f px, ξ, ξq dm λ px, ξq.
For g 1 P k pM q, let r g 1 be its G-invariant extension to Ă M , x r g 1 t pwq its Brownian motion on Ă M and m r g 1 its harmonic measure. The following limit exists almost surely:

lim tÑ`8 1 t b px,ξq ´xr g 1 t pwq ¯" ż M 0 ˆB Ă M ∆ r g 1 b px,ξq dm r g 1 ": 1 g 1 .
As g 1 Ñ g, m r g 1 Ñ m and hence both 1 g 1 , g 1 converges to . We may assume the neighborhood V g of g in Theorem 1.3 is such that :" min

g 1 PVg t g 1 , 1
g 1 u is positive. Consequently, for any curve λ Ñ g λ P V g , min λPp´1,1q

t g λ , 1 g λ u ě ą 0.
Lemma 3.9. Let V g be as in Theorem 1.3. For T ą 0 large enough, for all λ P p´1, 1q, x P M 0 and ξ, η

P B Ă M , ξ " η, 1 T E λ x,ξ ´pξ ˇˇηq x λ T ´pξ ˇˇηq x ¯ě 1 4 .
Proof. We may assume g λ is defined for λ P r´1, 1s. Assume the conclusion is not true. ´ˇp ξ ˇˇηq txts λ ´pξ ˇˇηq x ˇˇ¯ď 1 4 .

By using (3.3), (3.4) and suitably relabelling λ n , x n , ξ n and η n , we can find a sequence λ j P r´1, 1s, a sequence of integers N j Ñ 8, and points x j , ξ j and η j such that, for all j,

(3.5) 1 N j t 0 E λ j x j ,ξ j ˜pξ ˇˇηq x λ j N j t 0 ´pξ ˇˇηq x j ¸ă 1 2 .
By passing to suitable subsequences, we may also assume that λ n converges to some λ 0 P r´1, 1s, as n goes to infinity. For λ P r´1, 1s, write φ λ for the function on M 0 ˆB Ă M ˆB Ă M defined for x P M 0 and ξ, η P B Ă M , ξ " η, by

φ λ px, ξ, ηq " 1 t 0 E λ x,ξ ´pξ ˇˇηq x λ t 0 ´pξ ˇˇηq x ¯.
Then, by (3.2), φ λ has a continuous extension to the diagonal, still denoted φ λ , given by

φ λ px, ξ, ξq " 1 t 0 E λ x,ξ ´bpx,ξq px λ t 0 q ¯.
Write tx t s λ " β λ t x λ t , where β λ t P G and x λ t P M 0 . Using φ λ , (3.5) shows that there exist sequences λ j Ñ λ 0 , N j Ñ `8, as j Ñ 8, and points x j , ξ j , η j , such that for all j,

1 N j N j ´1 ÿ k"0 E λ j x j ,ξ j ´φpx λ j kt 0 , pβ λ j kt 0 q ´1ξ j , pβ λ j kt 0 q ´1η j q ¯ă 1 2 .
This means for λ j , N j , x j , ξ j and η j as above,

(3.6) 1 N j N j ´1 ÿ k"0 Q 2,λ j kt 0 φ λ j px j , ξ j , η j q ă 1 2 .
Define a sequence of probability measures µ j on M 0 ˆB Ă M ˆB Ă M by

µ j :" 1 N j N j ´1 ÿ k"0 pQ 2,λ j kt 0 q ˚pδpx j , ξ j , η j qq dp¨, ¨, ¨q,
where pQ 2,λ j kt 0 q ˚is the dual action of Q 2,λ j kt 0 and δpx j , ξ j , η j q is the Dirac measure at px j , ξ j , η j q. Then,

› › pQ 2,λ j t 0 q ˚µj ´µj › › ď 2 N j .
Moreover, pQ 2,λ j t 0 q ˚converges to pQ 2,λ 0 t 0 q ˚in norm as j goes to infinity by Theorem 1.3 since

› › Q 2,λ j t 0 ´Q2,λ 0 t 0 › › ď sup xP Ă M ˇˇˇż Ă M
p λ j pt, x, yq dVol λ j pyq ´pλ 0 pt, x, yq dVol λ 0 pyq ˇˇ"

sup xP Ă M ż λ j λ 0 ż Ă M
ˇˇpln p λ q p1q λ pt 0 , x, yq `pln ρ λ q p1q λ pyq ˇˇp λ pt 0 , x, yq dVol λ pyq dλ ďConst. |λ j ´λ0 | , where ρ λ " dVol λ {dVol 0 . Consequently, if µ is a weak limit of µ j , we have pQ 2,λ 0 t 0 q ˚µ " µ.

Let µ 1 " p1{t 0 q ş t 0 0 pQ 2,λ 0 s q ˚µ ds. The measure µ 1 is Q 2,λ 0 t -invariant (t ą 0) and hence coincides with m 2,λ 0 by Lemma 3.8. Note that φ λ j converges to φ λ 0 as j goes to infinity. We conclude from (3.6) that ş φ λ 0 dµ ď {2. Using (3.4) again, we find that

ż φ λ 0 dm 2,λ 0 ď 3 4 .
But, by Lemma 3.8, we also have

ż φ λ 0 dm 2,λ 0 " 1 t 0 ż E λ 0 x,ξ ´bx,ξ px λ 0 t 0 q ¯dm 2,λ 0 " lim tÑ8 1 t ż E λ 0 x,ξ ´bx,ξ px λ 0 t q ¯dm λ 0 ě ,
which is a contradiction.

Proof of Lemma 3.7. For λ P p´1, 1q, x P M 0 , ξ, η P B Ă M and t P R `, write

ψ λ b px, ξ, η, tq :" E λ x,ξ
ˆe´b `pξ|ηq tx t s λ ´pξ|ηqx ˘˙.

For each λ and b, it is true by the Markov property of the r g λ -Brownian motion that sup

x,ξ,η

ψ λ b px, ξ, η, t 1 `t2 q ď sup x,ξ,η ψ λ b px, ξ, η, t 1 q ¨sup x,ξ,η ψ λ b px, ξ, η, t 2 q.
Hence for Lemma 3.7, it suffices to find, for a fixed T and b 1 sufficiently small, positive numbers C 1 and k 1 such that for all λ P p´1, 1q and b ă b 1 , sup

x,ξ,η sup 0ďtăT ψ λ b px, ξ, η, tq ď C 1 , (3.7) sup x,ξ,η ψ λ b px, ξ, η, T q ď k 1 ă 1. (3.8)
Let T be as in Lemma 3.9. Note that there is some constant C such that ˇˇpξ|ηq txts λ ´pξ|ηq x ˇˇď 2dptx t s λ , xq ď Cd λ ptx t s λ , xq.

Using Taylor's expansion of the exponential function, we obtain e ´bppξ|ηq tx t s λ ´pξ|ηqxq ď 1 ´bppξ|ηq txts λ ´pξ|ηq x q `pCbd λ ptx t s λ , xqq 2 e Cbd λ ptxts λ ,xq .

Since the metrics r g λ have negative sectional curvatures bounded uniformly away from 0 for all λ, we have the exponential decay of the kernel functions, which implies that there exists some constant C 1 such that for all t, 0 ď t ď T , and all λ, E λ

x,ξ ´`Cd λ ptx t s λ , xq ˘2e Cd λ ptxts λ ,xq ¯ă C 1 . So, using Lemma 3.9, we obtain for b ď 1,

sup 0ďtăT ψ λ b px, ξ, η, tq ď 1 `bC 1 `b2 C 1 , ψ λ b px, ξ, η, T q ď 1 ´1 4 b `b2 C 1 .
Put b 1 " mint1, {p8C 1 qu. We see that (3.7) and (3.8) are satisfied for all λ P p´1, 1q and b ă b 1 with C 1 " 1 ` {8 ` 2 {p64C 1 q, k 1 " 1 ´ 2 {p64C 1 q.

Proof of Theorem 1.2. Let T ą 0 be fixed. Assume g P k pM q. By Proposition 3.6, there exist some neighborhood V g of g in k pM q such that for any continuous curve λ Þ Ñ g λ in V g , there is some positive b and k 0 ă 1 such that for all f P H 0 b , n P N,

(3.9) › › › › pQ λ T q n f ´ż f dm λ › › › › b ď k n 0 }f } b .
(For later consideration, we choose b to be small such that 2b also fulfills the requirement of Proposition 3.6 and 2b ă b 1 , where b 1 is from Lemma 3.7.) The inequality (3.9) means each operator Q λ T is a bounded operator on H 0 b , 1 is its isolated eigenvalue and m λ is the eigenfunction of eigenvalue 1 of the dual operator pQ λ T q ˚. By the classical spectrum theory on operators in Banach space (cf. [START_REF] Kato | Perturbation theory for linear operators[END_REF]Theorem 6.17]), we can decompose H 0 b into the direct sum of one-dimensional E ρ associated to the eigenvalue 1, and an infinitedimensional space E ă1 on which pQ λ T q n tends exponentially fast to 0. Let C be any circle around 1 with a small radius. Then the projection of f P H 0 b to E 1 is given by 1 2iπ

ż C ´zId ´Qλ T ¯´1 f dz.
Using this and (3.9), we conclude that the following two functional on H 0 b coincide:

ż ¨dm λ " 1 2iπ ż C ´zId ´Qλ T ¯´1
¨dz.

For the regularity of λ Þ Ñ m λ , we mean the regularity of λ Þ Ñ ş ¨mλ , which is the composition of two mappings

λ Þ Ñ Q λ T and Q λ T Þ Ñ 1 2iπ ż C ´zId ´Qλ T ¯´1
¨dz.

Note that by spectral continuity results for isolated simple eigenvalues (cf. [START_REF] Kato | Perturbation theory for linear operators[END_REF]Theorem 3.11]), for L P pH 0 b q ˚in a small neighborhood of Q 0 T , the mapping

L Þ Ñ 1 2iπ ż C pzId ´Lq ´1 ¨dz
is analytic. We may assume V g is such that all Q λ T belong to this neighborhood. Then for the regularity of λ Þ Ñ ş ¨mλ , it remains to show the regularity of the mapping

λ Þ Ñ Q λ T . For f P H 0 b , let r f be its G-invariant extension to Ă M ˆB Ă M . Then Q λ T f px, ξq " ż Ă M
r f py, ξqp λ pT, x, yq dVol λ pyq.

Put ρ λ :" dVol λ {dVol 0 . Then λ Þ Ñ ρ λ is C k in λ in C k p Ă M q
. By Theorem 1.3 i) and iii), for every i, 1 ď i ď k ´2, and every px, ξq P Ă M ˆB Ă M , the following differential exists:

`Qλ T f px, ξq ˘piq λ " i ÿ j"0
ˆi j ˙ż r f py, ξqpp λ q pjq λ pT, x, yqpρ λ q pi´jq λ pyq dVol 0 pyq.

To conclude this defines the i-th differential of Q λ T in λ in pH 0 b q ˚, we only need to show it defines a bounded operator from H 0 b into itself. For V g small, the norms of the differentials pln ρ λ q piq λ , i " 1, ¨¨¨, k ´2, and hence the norms of pρ λ q piq λ {ρ λ , i " 1, ¨¨¨, k ´2, are all bounded. So it suffices to consider S i λ , where

`Si λ f ˘px, ξq :" ż yP Ă M r f py, ξqpp λ q piq λ pT, x, yq dVol λ pyq,
and show it is a bounded functional of H 0 b . For each ξ P Ă M , r f p¨, ξq is uniformly continuous in x and bounded. Hence Theorem 1.3 iv) applies and shows that `Si λ f ˘px, ξq is continuous in x. Using Theorem 1.3 iii), we continue to compute that ˇˇ`S i λ f ˘px, ξq ˇˇď }f } 8 ¨ż pp λ q piq λ pT, x, yq p λ pT, x, yq p λ pT, x, yq dVol λ pyq ď c λ,piq p2q}f } b , where c λ,piq p2q is as in (1.5). For the Hölder continuity of ξ Þ Ñ `Si λ f ˘px, ξq and the corresponding Hölder norm estimation, it suffices to show the latter is bounded. By Hölder's inequality, Theorem 1.3 iii) and Lemma 3.7, we obtain

ˇˇ`S i λ f qpx, ξ 1 q ´`S i λ f qpx, ξ 2 q ˇˇe bpξ 1 ˇˇξ 2 qx ď ˆż Ă M ˇˇr f py, ξ 1 q ´r f py, ξ 2 q ˇˇ¨ˇˇpp λ q piq λ pT, x, yq ˇˇdVol λ pyq ˙ebpξ 1 ˇˇξ 2 qx ď }f } b ż Ă M e ´b´p ξ 1 ˇˇξ 2 qy´pξ 1 ˇˇξ 2 qx ¯ˇp p λ q piq λ pT, x, yq p λ pT, x, yq ˇˇˇˇp λ pT, x, yq dVol λ pyq " }f } b ¨´E λ x,ξ 1 ´e´2bppξ 1 |ξ 2 q tx T s λ ´pξ 1 |ξ 2 qxq ¯¯1 2 › › › › › pp λ q piq λ pT, x, yq p λ pT, x, yq › › › › › L 2 ď c λ,piq p2qpk T 1 q 1 2 }f } b .
Altogether, we have that each S i λ maps H 0 b into itself and is a bounded operator since

}S i λ f } b " sup x,ξ ˇˇ`S i λ f ˘px, ξq ˇˇ`sup x,ξ 1 ,ξ 2 ˇˇ`S i λ f qpx, ξ 1 q ´`S i λ f qpx, ξ 2 q ˇˇe bpξ 1 ˇˇξ 2 qx ď c λ,piq p2q `1 `pk T 1 q 1 2 ˘}f } b .
3.3. Differentials of the linear drift. We are in a situation to prove Theorem 1.1.

Proof of Theorem 1.1. It suffices to show the first statement.

Let V g be such that Proposition 3.5 and Theorem 1.2 hold true. We may also assume the Hölder exponents α of Proposition 3.5 and b of Theorem 1.2 coincide. As before, for any C k curve λ P p´1, 1q Þ Ñ g λ P V g with g 0 " g, we write X λ for the r g λ -geodesic spray, Div λ for the divergence operator associated with the r g λ -stable foliation and m λ for the g λ -harmonic measure on SM . Let λ be the linear drift of g λ . By (2.6),

(3.10) λ " ´żM 0 ˆB Ă M `Div λ X λ ˘px, ξq dm λ " ´Lλ `Div λ X λ ˘. By Proposition 3.5 iii), λ Þ Ñ Div λ X λ is C k´3 into C b p Ă M ˆB Ă M , R `q and is C k´2 into C 0 p Ă M ˆB Ă M , R `q. Write pDiv λ X λ q p0q λ " Div λ X λ and pDiv λ X λ q piq λ , i " 1, ¨¨¨, k ´2, for its i-th derivative in λ. Then pDiv λ X λ q piq λ belongs to C 0 p Ă M ˆB Ă M , R `q for i ď k ´2, and belongs to C b p Ă M ˆB Ă M , R `q for i ď k ´3. Regard each m λ as a measure on M 0 ˆB Ă M .
The operator L λ :" ş M 0 ˆB Ă M ¨dm λ is an bounded operator on continuous functions on M 0 ˆB Ă M . Moreover, by Theorem 1.2, λ Þ Ñ L λ is C k´2 differentiable as elements of pH 0 b q ˚. Using these regularities and (3.10), we conclude that the function λ Þ Ñ λ is C k´2 differentiable. Denote by L piq λ , i " 1, ¨¨¨, k ´2, the i-th differential functional of L λ . Then, for every i, 1 ď i ď k ´2, the i-th differential of λ in λ, i.e., piq λ , is given by

(3.11) piq λ " ´i ÿ j"0 ˆi j ˙Lpjq λ ´pDiv λ X λ q pi´jq λ ¯.
Specifying (3.11) for i " 1, we write:

Corollary 3.10. Let g P 3 pM q. For any C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q with g 0 " g and constant volume, we have

(3.12) p λ q 1 0 " ´ż Div 0 X 0 dpm λ q 1 0 ´ż `Div λ X λ ˘1 0 dm 0 .
In particular, if g " g 0 is a locally symmetric metric and the volume Vol λ pM q is constant in λ, then we have p λ q 1 0 " 0.

Proof. We apply (3.11) for i " 1 and λ " 0. The operator L 0 extends to the harmonic measure m 0 and L p1q 0 is a linear functional on the space H 0 b that we denote pm λ q 1 0 . Formula (3.12) follows.

Let υ λ be the volume entropy of p Ă M , r g λ q,

υ λ :" lim RÑ8 1 R ln Vol λ Bpx, Rq,
where Bpx, Rq is the ball of radius R about x in Ă M . We know by [START_REF] Kaimanovich | Brownian motion and harmonic functions on covering manifolds. An entropic approach[END_REF] that for all λ, λ ď υ λ and by [BCG] that the volume entropy of a negatively curved locally symmetric space achieves its minimum over all metrics of the same volume on that space. Since λ Þ Ñ λ and λ Þ Ñ υ λ are differentiable at 0 (by Theorem 1.1 and by [KKPW]), the derivative has to be 0.

We develop formula (3.12). The vector pX λ q 1 0 is a vertical vector given by [LS2, Proposition 4.5]. For v " px, ξq, it is the sum of ´}X λ } r g 0 ¯1 0 pvqX 0 pvq and of a vector Y pvq orthogonal to v, where Y is a C 1 vector field along the stable manifolds. Let u 0 be the function such that ∆u 0 " ´DivX ´ , psee [LS2, (5.12)]q.

Theorem 3.11. Let M be a closed connected smooth manifold and let g P 3 pM q. For any C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q with g 0 " g and constant volume,

p λ q 1 0 " ż ˆ´1 2 x∇traceX , Xy `1 2 X pX, XqDivpXq `1 2 x∇pX pX, Xqq, Xy ´DivY ˙dm `ż ˆ´1 2 x∇traceX , ∇u 0 y `Div `X p∇u 0 q ˘˙dm, (3.13)
where we omit the index 0 for ∇ 0 , X 0 , x¨, ¨y0 , Div 0 and m 0 at g 0 , and where X p¨q is considered as the p1, 1q-form in Ă M such that xX pZq, Z 1 y " X pZ, Z 1 q. In particular, there is a linear functional L on C k pS 2 T ˚q such that p λ q 1 0 " LpX q.

Proof. To obtain (3.13), we use the decomposition of p λ q 1 0 given by (3.12) as above:

p λ q 1 0 " ´ż pDiv λ Xq 1 0 dm ´ż pDivX λ q 1 0 dm ´ż DivX dpm λ q 1 0 and study the three terms successively.

Firstly, we have pDiv λ Xq 1 0 " 1 2 x∇pTraceX q, Xy.

Then, for v " px, ξq, pX

λ q 1 0 is the sum of ´}X λ } ¯1 0 pvqXpvq and Y pvq. Hence, pDivX λ q 1 0 " DivY `Div ˆ´}X λ } ¯1 0 pvqXpvq ˙. Since }X λ } 2 g λ " 1, we have ´}X λ } ¯1 0 pvq " ´1 2 X pXpvq, Xpvqq.
Thus,

Div ˆ´}X λ } ¯1 0 pvqXpvq ˙" ´1 2 X pXpvq, XpvqqDivpXpvqq ´1 2 x∇pX pXpvq, Xpvqqq, Xpvqy.
Lastly, we discuss the term ş DivX dpm λ q 1 0 . Recall that, by Theorem 1.2, λ Þ Ñ m λ is differentiable at 0, with derivative pm λ q 1 0 P pH 0 b q ˚(denoted as an integral). It follows that, for f smooth on SM , (3.14)

ż p∆ λ q 1 0 f dm `ż ∆f dpm λ q 1 0 " 0.
The equation (3.14) extends to functions f that are of class C 2 along the stable leaves with globally continuous second order derivatives. In particular, (3.14) applies to the function u 0 and therefore, ż DivX dpm λ q 1 0 " ż p∆ λ q 1 0 u 0 dm " ż ˆ1 2 x∇u 0 , ∇traceX y ´Div `X p∇u 0 q ˘˙dm.

To show p λ q 1 0 is linear in X , it remains to consider ş DivY dm. If we denote kpx, y, ξq the continuous version of the density `dm y {dm x ˘pξq (see e.g. [LS2, Proposition 2.2]), the integration by parts formula yields (3.15)

ż
DivY dm " ´ż xY, ∇ y ln kpx, y, ξq| y"x y dm.

We recall from [START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF]Proposition 4.5] the construction of the vector field Y . Let v P T M . We define the vector Υpvq P T T M as the vertical vector with vertical component given by

Υpvq :" `∇λ v pvq ˘1 0 ´@`∇ λ v pvq ˘1 0 , v D .
Clearly, for all v P SM , Υpvq depends linearly on X , and sup v }Υpvq} is bounded by C}X } C 1 . In order to obtain Y pvq, we consider the orbit Φ s pvq, s ě 0, under the geodesic flow. For each s ě 0, we decompose ΥpΦ s pvqq into a sum of its unstable part ΥpΦ s pvqq u and its stable part. The vector Y pvq is the vertical part of ż 8 0 pDΦ s q ´1ΥpΦ s pvqq u ds.

Since the geodesic flow is Anosov, there are C, τ ą 0 such that pDΦ s q ´1 restricted to the unstable manifold has norm smaller than Ce ´τ s . It follows that the expression ş DivY dm is linear in X and bounded by C}X } C 1 .

Remark 3.12. We can also verify that the formula (3.13) gives indeed 0 in the case when g " g 0 is locally symmetric.

Assume that g is a locally symmetric metric, then DivX is the constant ´ and the measure m is the normalized Liouville measure. Since the measures m λ are normalized (and the constant functions belong to the space H 0 b ), ş DivX dpm λ q 1 0 " 0 and formula (3.13) reduces to

p λ q 1 0 " ż ˆ´1 2 x∇traceX , Xy `1 2 X pX, XqDivpXq `1 2 x∇pX pX, Xqq, Xy ´DivY ˙dm ":
ż ppIq `pIIq `pIIIq `pIVqq dm.

Since traceX , X pX, Xq are functions on SM and we integrate with respect to the invariant Liouville measure, the integrals of pIq, pIIIq vanish. Since r g is a symmetric space, the kpx, y, ξq in formula (3.15) is given by ´ 0 b px,ξq pyq, where b px,ξq is the Busemann function (see Section 2.4). It follows that ∇ y ln kpx, y, ξq| y"x " Xpvq. Since Y pvq is orthogonal to Xpvq, the integral ş pIVq dm vanishes as well. Remains to consider

ż pIIq dm " ´1 2 ż X pX, Xq dm " ´1 2 m ż M 0 traceX dVol r g Vol r g pM 0 q " ´ m `Vol r g λ pM 0 q ˘1 0 Vol r g pM 0 q
, where Vol r g is the Riemannian volume. So, ş pIIq dm vanishes since the volume is constant.

Brownian motion and stochastic flows

In this section, we recall the Eells-Elworthy-Malliavin construction of the Brownian motion on a manifold through a stochastic differential equation (SDE) on the orthogonal frame bundle and of the associated stochastic flow (see Proposition 4.27). We give estimations on the growth in time of the derivatives of this stochastic flow. We will need in Sections 5 and 6 both uniform estimations and estimations in average with respect to the Brownian motion and Brownian bridge distributions in the non-compact case.

4.1. Parallelism and the Brownian motion. Let N be a C 8 n-dimensional Riemannian manifold. A differential form ϑ on N with values in R n is called a parallelism differential form [START_REF] Malliavin | Stochastic Jacobi fields, Partial differential equations and geometry[END_REF]), if it realizes for every u P N an isomorphism of T u N on R n . A parallelism differential form ϑ is called C k if it is a C k section of the frame bundle space FpNq of N.

Let f : r0, `8q Ñ R n be a C 2 curve. It defines a one parameter family of continuous vectors tpdf {dtq| t"τ u τ Pr0,`8q . Let ϑ be a C 1 parallelism differential form. It, together with f , defines a C 1 vector field on N ˆR`:

Z f t,u :" ϑ ´1 u p df dt q, @u P N, t P R `.
By the classical theory of ordinary differential equation, there exists a flow F f,t generated by Z f t,u , which solves Cauchy's problem d dt pF f,t pu 0 qq " Z f t,uptq , where uptq " F f,t pu 0 q and F f,0 pu 0 q " u 0 P N.

The orbit of each u 0 P N under F f,t is an analogue of the curve f since the velocity at time τ is just the preimage of pdf {dtq| t"τ by ϑ. Moreover, the time t map F f,t depends C 1 on the initial point u 0 . The variation of F f,t pu 0 q with respect to u 0 reflects the geometric difference between N and R n and the pull back of the tangent map of F f,t in R n via ϑ can be formulated using the equation of dϑ ([Mal2, Proposition 3.2]). In general, if f is a C k`1 curve in R n and ϑ is C k , then the flow generated by Z f t,u depends C k on the initial point.

In case N is the frame bundle space of Ă M , there are plenty of parallelism differential forms using the dual form and the connection forms. Recall that a frame u for T x Ă M , x P Ă M , is an ordered basis u " pu 1 , ¨¨¨, u m q for T x Ă M , which defines a linear isomorphism form R m to T x Ă M by letting upyq :" ř m i"1 y i u i , for y " py i q P R m . The set of all frames u for all tangent spaces T x Ă M , denoted by Fp Ă M q, is a C 8 manifold. The dual form (or the canonical form) on Fp Ă M q is an R m -valued 1-form defined by θ u pY q :" u ´1pπ ˚Y q, @Y P T u Fp Ă M q, where π ˚is the tangent map of the natural projection map from Fp Ă M q to Ă M . The kernel of π ˚is the vertical vector bundle of T Fp Ă M q:

V T Fp Ă M q :" Y P T Fp Ă M q : π ˚Y " 0 ( .

For A P glpm, Rq, let A ˚be the vector field on Fp Ă M q with A ˚puq " 9 γ u p0q, where γ u ptq " R expptAq u and R a denotes the right action by a. A C k (Ehresmann) affine connection for pFp Ă M q, π, ¨q is a C k glpm, Rq-valued 1-form on Fp Ă M q satisfying pA ˚puqq " A, @A P glpm, Rq, ppR a q ˚Y q " Adpa ´1q pY q, @a P GLpm, Rq, Y P T Fp Ă M q.

Each C k affine connection form of Fp Ă M q assigns a unique C k -distributed complementary horizontal vector bundle HT Fp Ă M q, the kernel of , which is invariant under the right action of GLpm, Rq. Each induces the notion of covariant derivative ∇, D on vector fields and forms on Fp Ă M q, respectively. Let T, R be the corresponding torsion tensor and curvature tensor, and Θ :" Dθ, Ω :" D be the torsion form and curvature form. Then

T pX, Y q " upΘp X, Y qq, RpX, Y qZ " upΩp X, Y q ¨pu ´1Zqq,
where X, Y , Z P T u Fp Ă M q are any vectors which project to X, Y, Z P T x Ă M , respectively, and u P F x pM q can be chosen arbitrarily. Any pair pθ, q is a parallelism differential form for Fp Ă M q. It satisfies the following structure equations (cf. [START_REF] Spivak | A comprehensive introduction to differential geometry[END_REF]p. 327]):

dθpY 1 , Y 2 q " ´t pY 1 q ¨θpY 2 q ´ pY 2 q ¨θpY 1 qu `ΘpY 1 , Y 2 q, (4.1) d pY 1 , Y 2 q " ´r pY 1 q, pY 2 qs `ΩpY 1 , Y 2 q, (4.2)

where Y 1 , Y 2 P T u Fp Ă M q and pY 1 q ¨θpY 2 q is the action of the matrix pY 1 q on θpY 2 q P R m .

For g P M k pM q, let O r g p Ă M q Ă Fp Ă M q be the collection of r g-orthogonal frames, the so-called orthogonal frame bundle space of p Ă M , r gq. Each u P O r g x p Ă M q defines an isometry from R m with the classical Euclidean metric to pT x Ă M , r gq. Let be the unique torsion free connection form on Fp Ă M q which induces the r g-connection ∇ and curvature tensor R. Then " p i j q, Ω " pΩ i j q satisfy i j "

ÿ k Γ i kj θ k , Ω i j " 1 2 ÿ k,l R i jkl θ k ^θl ,
where Γ and R are ∇ and R read in the frame u. The structural equations (4.1) and (4.2) of pθ, q are reduced to

dθ i pY 1 , Y 2 q "
´` i j pY 1 qθ j pY 2 q ´ i j pY 2 qθ j pY 1 q ˘, (4.3) d i j pY 1 , Y 2 q " ´´ i q pY 1 q q j pY 2 q ´ i q pY 2 q q j pY 1 q ¯`R i jkl θ k pY 1 qθ l pY 2 q, (4.4)

where Y 1 , Y 2 P T u Fp Ă M q and u P Fp Ă M q. The restriction of pθ, q to O r g p Ă M q also defines a parallelism differential form. For instance, we can use this parallelism to recover the geodesic flow on S Ă M . Let f : r0, `8q Ñ OpR m q be a half line with df {dt " p e, 0q for some unit vector e P R m . It defines a C k´1 vector field on O r g p Ă M q ˆR`b y letting

Z f t,u :" pθ, q ´1 u p df dt q, @u P O r g p Ă M q,
where each Z f t,u is just the lift of ue to HT Fp Ă M q. Let F e,t denote the flow generated by Z f t,u with df {dt " pe, 0q. It projects to the r g-geodesic flow on S Ă M and the orbit of u P O r g p Ă M q under it is the parallel transportation of u along the unit speed geodesic γ ue .

The key point of the Eells-Elworthy-Malliavin construction of the Brownian motion on a Riemannian manifold is to realize it as a transportation of the R m -Brownian motion using the parallelism differential form of the orthogonal frame bundle.

Let Θ `be the space of continuous paths w : r0, `8q Ñ R m , equipped with the smallest σ-algebra F for which the projections R t : w Þ Ñ wptq are measurable. The sub σ-algebras tF t u tPR `of F is an increasing sequence such that tR s u sďt are measurable in F t . An R m -Brownian motion is a continuous time random process tB t : B t pwq " wptqu tPR `on Θ ẁith distribution Q so that the induced actions Q t : pQ t ϕqpxq " E x pϕpB t pwqqq on smooth functions ϕ form a semigroup with Euclidean Laplacian ∆ Eu as being the infinitesimal generator (lim tÑ0 pQ t ϕ´ϕq{t " ∆ Eu ϕ whenever ϕ P C 2 c pR m q, the collection of C 2 functions on R m with compact support). In other words, (4.5) B t " pB 1 t , ¨¨¨, B m t q, where all B i t are independent 1-dimensional Brownian motions on R with time t transition probability p4πtq ´1 2 e ´px i ´yi q 2 4t between points x i and y i in R. In the language of Stratonovich stochastic differential equation (SDE), (4.5) is

dB t " m ÿ i"1 e i pB t q ˝dB i t ,
where te i " B{Bx i u is an orthogonal chart of R m , which means for all ϕ P C 8 c pR m q, the collection of C 8 functions on R m with compact support, and for all t P R `, ϕpB t q " ϕpB 0 q `ż t 0 m ÿ i"1 e i ϕpB s q ˝dB i s .

Fix a C 8 function c, with support contained in the unit interval r0, 1s with integral 1. For each ą 0, let c pτ q :" ´1cp ´1τ q be an approximate unit function. For any sample path t Þ Ñ wptq " pw 1 ptq, ¨¨¨, w m ptqq of B, we can smooth it using c by letting [START_REF] Malliavin | Stochastic Jacobi fields, Partial differential equations and geometry[END_REF]).

w i ptq :" ż 0 w i pt `sqc
Given a sample path w of B t starting from the origin, the smoothed curve w has its lift in OpR m q with tangent vectors pdw {dt, 0q. Let g P M k pM q and let θ, and H be the associated dual form, r g-connection form and horizontal lift map, respectively. Consider the C k´1 vector field on O r g p Ă M q ˆR`:

Z f, t,u :" pθ, q ´1 u p dw dt , 0q, @u P O r g p Ă M q.
We see that

Z f, t,u " m ÿ i"1
Hpu, e i q ¨dw i dt ,

where Hpu, e i q is horizontal lift of ue i to HT Fp Ă M q. Let Φ f,t be the flow generated by Z f, t,u . For u P O r g p Ă M q, its orbit u ptq under Φ f,t solves the differential equation

(4.6) du ptq dt " m ÿ i"1
Hpu ptq, e i q ¨dw i dt .

The projection of the orbit t Þ Ñ u ptq to Ă M has tangent u ptqpdw {dtq at time t and is an analog of the curve w . As w varies, the distribution of the projection of u ptq on Ă M simulates the distribution of the R m Brownian motion. As tends to 0, almost surely, the differential system (4.6) tends to (4.7)

du t " m ÿ i"1
Hpu t , e i q ˝dB i t pwq, which means for all smooth function ϕ on O r g p Ă M q,

ϕpu t q " ϕpu 0 q `ż t 0 m ÿ i"1
pHpu s , e i qϕqpu s q ˝dB i s , 0 ď t ă 8.

Since the vector fields Hp¨, e i q are C k´1 , for any initial u 0 , there exists a unique solution u " pu t q tPR `to (4.7), which is continuous in pt, u 0 q for all t P R `(see Proposition 4.1).

Recall that the generator A of u t is such that ϕpu t q ´ϕpu 0 q ´ż t 0 Aϕpu s q ds is a local martingale for all smooth ϕ. By Itô's formula, we see that

A " m ÿ i"1
Hp¨, e i q 2 , which is the Bochner horizontal Laplacian ∆ O r g p Ă M q . It is a lift of the Laplacian ∆ in the sense that for any smooth function ϕ on Ă M and its lift ϕ to O r g p Ă M q, (4.8) ∆ O r g p Ă M q ϕpuq " ∆ϕpπuq.

Let x " px t q tPR `be the projection on Ă M of the solution u " pu t q tPR `of (4.7) with initial value u 0 P O r g x 0 p Ă M q. It defines a measurable map from orbits in Θ `starting from the origin to C x 0 pR `, Ă M q, the space of continuous paths on Ă M starting from x 0 . As x 0 varies, Qpx ´1q gives a distribution in the space of continuous paths on Ă M . For τ P R `, let C x 0 pr0, τ s, Ă M q be the collection of continuous paths ρ : r0, τ s Ñ Ă M with ρp0q " x 0 . Then x also induces a measurable map x r0,τ s : Θ `Ñ C x 0 pr0, τ s, Ă M q sending w to px t pwqq tPr0,τ s . So,

P τ :" Qpx ´1 r0,τ s q
gives the distribution probability of paths xpwq on Ă M up to time τ and this distribution is independent of the choice of the initial orthogonal frame u 0 that projects to x 0 . Since x has generator ∆ by (4.8), it visualizes the Brownian motion on Ă M . This is the Eells-Elworthy-Malliavin's approach to obtain the Brownian motion on a manifold (cf. [Elw]).

4.2.

A stochastic analogue of the geodesic flow. The regularity of the Brownian companion process u t with respect to its initials u 0 can be understood by general theory on stochastic flows associated to SDEs.

Let X 1 , ¨¨¨, X d be bounded vector fields on a smooth finite dimensional Riemannian manifold pN, x¨, ¨yq. Let pz t q tPR `" pz 1 t , ¨¨¨, z d t q be a continuous stochastic process on R d . An N-valued semimartingale px t q tPR `defined up to a stopping time τ is said to be a solution of the following Stratonovich SDE (4.9)

dx t " d ÿ i"1 X i px t q ˝dz i t ,
if for all ψ P C 8 pNq,

ψpx t q " ψpx 0 q `ż t 0 d ÿ i"1 X i ψpx s q ˝dz i s , 0 ď t ă τ.
The solution to (4.9) always exists when all X i are C 1 bounded ( [Elw]). Note that X " pX 1 , ¨¨¨, X d q is a linear isomorphism from R d to T N. So, x t is a parallel transportation of z t to the manifold N via X. The pair pX, pz t q tPR `q is called a stochastic dynamical system (SDS) on N ( [Elw]) and it is said to be C j if all X i are C j bounded. Using X, we also write (4.9) as dx t " Xpx t q ˝dz t . The mapping F t p¨, wq : x 0 pwq Þ Ñ x t pwq has the following regularity with respect to the starting point x 0 pwq. Proposition 4.1. ([Elw, Theorem 3, Chapter VIII]) Let pX, pz t q tPR `q be a C j SDS on N. There is a version of the explosion time map x Þ Ñ τ x , defined for x P N, and a version of F t px, wq, defined when t P r0, τ x pwqq, such that if Npt, wq " tx P N : t ă τ x u, then the following are true for each pt,

wq P R `ˆΘ `. i) The set Npt, wq is open in N. ii) The map F t px, wq : Npt, wq Ñ N is C j´1 and is a diffeomorphism onto an open subset of N. Moreover, the map τ Þ Ñ F τ p¨, wq of r0, ts into C j´1 mappings of Npt, wq is continuous. Corollary 4.2. Let g P M k pM q (k ě 3).
There is a version of the solution flow

F t p¨, wq : u 0 pwq Ñ u t pwq, t P R `, to (4.7) in Fp Ă M q, which is a C k´2 diffeomorphism into Fp Ă M q and is continuous in t.
Proof. Each x P Ă M has infinite distance to the boundary. Hence each solution process u to (4.7) with u 0 P F x p Ă M q projects to be a diffusion process on Ă M starting from x and has infinity explosion time. Since g P M k pM q, the vector fields Hp¨, e i q, i " 1, 2, ¨¨¨, m, on O r g p Ă M q are all C k´1 bounded with respect to the r g metric. So F t p¨, wq : u 0 pwq Þ Ñ u t pwq is C k´2 with respect to the initial points u 0 and is continuous in t by Proposition 4.1.

For l ď j ´1, the l-th tangent map of F t in Proposition 4.1, denoted by D plq F t p¨, wq, can be formulated and its norm can be estimated if N is equipped with a reference connection.

Proposition 4.3. ( [Elw]) Let pX, pz t q tPR `q be a C j SDS on N. Assume there is a Levi-Civita connection ∇ induced by some metric such that the covariant derivatives ∇ ι X i , ι " 0, 1, ¨¨¨, j, i " 1, ¨¨¨, d, are bounded and the curvature tensor R of ∇ and its first j ´1 derivatives are bounded. The following hold true.

i) There is a version of tF t p¨, wqu such that almost surely, for l ď j ´1, t P R `and v 0 pwq P T plq N, v t pwq :" rD plq F t p¨, wqsv 0 pwq satisfies the Stratonovich SDE

dv t " d ÿ i"1 rD plq X i px t qsv t ˝dz i t ,
where, if we denote by F i t the deterministic flow map generated by the vector field X i and D plq F i t its l-th differential map, then for v P T plq N with footpoint x P N,

rD plq X i pxqsv :" D dt prD plq F i t svq.
ii) For any q P r1, 8q, there is a bounded function c l pt, qq, which depends on t, m, q, and the bounds of ∇ ι X i and ∇ ι´1 R, ι ď l `1, such that }rD plq F t p¨, wqs} L q ă c l pt, qq.

Proposition 4.3 applies to the flow map corresponding to (4.7). So we can formulate the SDEs of trD plq F t p¨, wqsu. We will use them to specify c l pt, qq and give a more detailed study of their norm growths in time for later use.

Let F t p¨, wq be as in Corollary 4.2. The first order tangent map D p1q F t pu 0 , wq records the first order infinitesimal response of F t pu 0 , wq to the change of initial point u 0 . Let

C : p´1, 1q Þ Ñ Fp Ă M q be a differential curve with Cp0q " u 0 , C 1 p0q " v. Then v t :" " D p1q F t pu 0 , wq ‰ v " D Bs F t pCpsq, wq ˇˇˇs "0
.

The SDEs of v t can be formulated using the parallelism form pθ, q as follows.

Lemma 4.4. ([Mal2, Theorem 5.1]) Let F t p¨, wq be as in Corollary 4.2.

i) For any v P T u 0 Fp Ă M q, v t satisfies the Statonovich SDE dv t pwq " ∇pv t pwqqHpu t , ˝dB t q.

ii) Consider the map r Č D p1q F t pu 0 , wqs :" pθ, q ut ˝rD p1q F t pu 0 , wqs ˝pθ, q ´1 u 0 .

For pzp0q, zp0qq :" pz i p0q, z l j p0qq P T FpR m q, pzptq, zptqq :" r Č D p1q F t pu 0 , wqspzp0q, zp0qq satisfies the Stratonovich SDE " dzptq " zptq ˝dB t pwq, dzptq " u ´1 t R pu t ˝dB t pwq, u t zptqq u t . (4.10)

iii) The Itô form of (4.10) is

$ & % dzptq " zptq dB t pwq `Ricpu t zptqq dt, dzptq " u ´1 t R pu t dB t pwq, u t zptqq u t `u´1 t R pu t e i , u t zptqe i q u t dt `u´1
t p∇pu t e i qRq pu t e i , u t zptqq u t dt,

(4.11)
where the summation Σ m i"1 is omitted in (4.11) for simplicity and Ricpuzq :"

m ÿ i"1 u ´1Rpue i , uzque i , @u P Fp Ă M q, z P R m . (4.12)
For t, t, 0 ď t ă t ď T , let F t,t p¨, wq be the flow map of (4.7) sending u t to u t . Then F t pu 0 , wq " F 0,t pu 0 , wq " F t,t pu t , wq ˝F0,t pu 0 , wq.

Let rD plq F t,t p¨, wqs (l ď k ´2) be the l-th tangent map of F t,t . When l " 1, let r Č D p1q F t,t pu 0 , wqs :" pθ, q u t ˝rD p1q F t,t pu 0 , wqs ˝pθ, q ´1 ut .

Then rD p1q F t,t s (resp. r Č D p1q F t,t s) satisfies the same SDE as rD p1q F 0,t s (resp. r Č D p1q F 0,t s).

To describe " D p2q F t pu 0 , wq ‰ , we can follow [Elw] to use the horizontal/vertical Whitney sum decomposition of T pu,vq T Fp Ă M q " T u Fp Ă M q ˆTu Fp Ă M q with respect to the Levi-Civita connection. The second order tangent vector pu, v; V 0 , V 1 q P T pu,vq T N is in one-to-one correspondence with the Jacobi field Y psq along the geodesic s Þ Ñ Cpsq :" exppsvq with Y p0q " V 0 , ∇Y p0q " V 1 , where Y p0q tells the infinitesimal change of Cp0q (i.e., the horizontal part change of pCp0q, C 1 p0qq) and ∇Y p0q tells the the infinitesimal change of C 1 p0q along the geodesic from u 0 with initial velocity V 0 (i.e., the vertical part change of pCp0q, C 1 p0qq. For the geodesic τ Þ Ñ C 1 pτ q :" exppτ V 1 q, let v pτ q be the parallel transportation of v along C 1 to the point C 1 pτ q and define

∇ V 0 " D p1q F t pu 0 , wq ‰ pvq " D Bτ " D p1q F t pC 1 pτ q, wq ‰ pv pτ qq ˇˇˇτ "0 . (4.13)
Then for almost all w,

" D p2q F t pu 0 , wq ‰ pu 0 , v; V 0 , V 1 q " ´"D p1q F t pu 0 , wq ‰ pu 0 ; vq; " D p2q F t pu 0 , wq ‰`V 0 , V 1 ˘¯, where " D p2q F t pu 0 , wq ‰`V 0 , V 1 " ´"D p1q F t pu 0 , wq ‰ pV 0 q, ∇ V 0 " D p1q F t pu 0 , wq ‰ pvq `"D p1q F t pu 0 , wq ‰ pV 1 q ¯.
By Lemma 4.4, to describe " D p2q F t p¨, wq ‰ pV 0 , V 1 q, it remains to identify

V t pv, V 0 , wq :" ∇ V 0 " D p1q F t pu 0 , wq ‰ pvq.
Lemma 4.5. ([Elw, Lemma 5B, Chapter VIII]) Let g P M k pM q, k ě 4. For v P T u 0 Fp Ă M q, pV 0 , 0q P T pu 0 ,vq T Fp Ă M q, let v t :" rD p1q F t pu 0 , wqsv, V t :" rD p1q F t pu 0 , wqsV 0 .

i) On T Fp Ă M q, the process V t :" V t pv, V 0 , wq satisfies the Stratonovich SDE dV t " ∇pV t qHpu t , ˝dB t q `∇p2q pv t , V t qHpu t , ˝dB t q `RpHpu t , ˝dB t q, V t qv t .

ii) On T FpR m q, the process pθ, q ut pV t q satisfies the Stratonovich SDE d ppθ, q ut pV t qq " ` pV t q ˝dB t pwq, u ´1 t Rpu t ˝dB t , θpV t qqu t pθ,

q ut ´∇p2q pv t , V t qHpu t , ˝dB t pwqq `RpHpu t , ˝dB t pwqq, V t qv t ¯. (4.14)
iii) The Itô form of (4.14) is dθpV t q " pV t qdB t pwq `Ricpu t θpV t qqdt `Φθ pv t , V t , dB t , dtq, (4.15)

d pV t q " u ´1 t R pu t dB t pwq, u t θpV t qq u t `u´1 t R pu t e i , u t pV t qe i q u t dt (4.16) `u´1 t `∇pu t e i qR ˘pu t e i , u t θpV t qq u t dt `Φ pv t , V t , dB t , dtq,
where the summation Σ m i"1 is omitted and

Φ θ pv t , V t , dB t , dtq :"θ ´∇p2q pv t , V t qHpu t , dB t pwqq `RpHpu t , dB t pwqq, V t qv t 2 ´∇p2q pv t , V t qHpu t , e i q `RpHpu t , e i q, V t qv t ¯ei dt `θ
´"Hpu t , e i q, ∇ p2q pv t , V t qHpu t , e i q `RpHpu t , e i q, V t qv t ı¯d t,

Φ pv t , V t , dB t , dtq :" ´∇p2q pv t , V t qHpu t , dB t pwqq `RpHpu t , dB t pwqq, V t qv t 2u ´1 t R ´ut e i , u t θ `∇p2q pv t , V t qHpu t , e i q `RpHpu t , e i q, V t qv t ˘¯u t dt
`

´"Hpu t , e i q, ∇ p2q pv t , V t qHpu t , e i q `RpHpu t , e i q, V t qv t ı¯d t.

A corollary of Lemma 4.5 is that we can describe V t (resp. pθ, q ut pV t q) using the tangent

maps " D p1q F t,t pu 0 , wq ‰ (resp. " Č D p1q F t,t pu 0 , wq ‰
) by a stochastic version of the variation of constant method, i.e., a stochastic Duhamel principle.

Corollary 4.6. Let g P M k pM q (k ě 4) and let v t , V t and V t be as in Lemma 4.5.

iq V t " ż t 0 " D p1q F τ,t pu τ , wq ‰ ´∇p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ¯˝dB i τ .
iiq pθ, q ut pV t q " ż t 0 " Č D p1q F τ,t pu τ , wq ‰ pθ, q uτ ´∇p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ¯˝dB i τ . (4.17)

iii) The Itô form of (4.17) is pθ, q ut pV t q "

ż t 0 " Č D p1q F τ,t pu τ , wq ‰ ´r Φ θ pv τ , V τ , dB τ , dτ q, r Φ pv τ , V τ , dB τ , dτ q ¯, (4.18) where r Φ θ pv τ , V τ , dB τ , dτ q " Φ θ pv τ , V τ , dB τ , dτ q ´ ´∇p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ¯ei dτ, r Φ pv τ , V τ , dB τ , dτ q " Φ pv τ , V τ , dB τ , dτ q ´u´1 τ R ´uτ e i , u τ θ `∇p2q pv τ , V τ qHpu τ , e i q`RpHpu τ , e i q, V τ qv τ ˘¯u τ dτ.
Proof. For i) and ii), it suffices to show i) since it implies ii) by applying the pθ, q map. Regard the tangent map " D p1q F t pu 0 , wq ‰ as a random matrix solution y t pwq to dy t pwq " ∇py t pwqqHpu t , ˝dB t q, y 0 " Id.

Put

υ t :"V 0 `ż t 0 " D p1q F τ pu 0 , wq ‰ ´1 ´∇p2q pv τ , V τ qHpu τ , ˝dB τ q `RpHpu τ , ˝dB τ q, V τ qv τ ¯.
Then the differentiation rule of Stratonovich integral shows that dpy t υ t q " p˝dy t qυ t `yt ˝dυ t " ∇py t pwqυ t qHpu t , ˝dB t q `∇p2q pv t , V t qHpu t , ˝dB t q `RpHpu t , ˝dB t q, V t qv t , where d should be understood as the covariant derivative. Since y 0 υ 0 " V 0 " 0, we obtain

V t " y t υ t " ż t 0 " D p1q F τ,t pu τ , wq ‰ ´∇p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ¯˝dB i τ .
Regard " Č D p1q F t p¨, wq ‰ as a matrix solution y t pwq to (4.11) with y 0 " Id. Put

r υ t :" V 0 `ż t 0 " Č D p1q F τ pu 0 , wq ‰ ´1 ´r Φ θ pv τ , V τ , dB τ , dτ q, r Φ pv τ , V τ , dB τ , dτ q ¯.
Write y t υ t :" ppy t υ t q θ , py t υ t q q, where py t υ t q θ P R m and py t υ t q P FpR m q. Then the Itô form infinitesimal differentiation rule shows that dpy t r υ t q " pdy t qr υ t `yt dr υ t `dy t ¨dr υ t " `py t r υ t q dB t pwq `Ricpu t py t r υ t q θ qdt `Φθ pv t , V t , dB t , dtq,

u ´1 t R pu t dB t pwq, u t py t r υ t q θ q u t `u´1 t R pu t e i , u t py t r υ t q e i q u t dt `u´1 t ∇Rpu t e i q pu t e i , u t py t r υ t q θ q u t dt `Φ pv t , V t , dB t , dtq ˘.
This means y t r υ t with r υ 0 " p0, 0q solves (4.15) and (4.16). Thus (4.18) holds true.

For " D p2q F t pu 0 , wq ‰ on T pu,vq T u Fp Ă M q " T u Fp Ă M q ˆTu Fp Ă M q, we can define its Euclidean companion map " Č D p2q F t pu 0 , wq ‰ on T FpR m q ˆT FpR m q as follows. For `V0 , V 1 ˘P T FpR m q ˆT FpR m q, let pV 0 , V 1 q :" pppθ, q u 0 q ´1pV 0 q, ppθ, q u 0 q ´1pV 1 qq. Let V i,t :" rD p1q F t pu 0 , wqsV i for i " 0, 1 and let v t , V t be defined as in Lemma 4.5. Then

" Č D p2q F t pu 0 , wq ‰`V 0 , V 1 ˘:" `pθ, qpV 0,t q, pθ, qpV 1,t `Vt q ˘.
We can continue the above discussion to formulate

" D plq F t p¨, wq ‰ , 3 ď l ď k ´2. Put # `up2q ; v p2q ˘": pu, v; V 0 , V 1 q, `uplq ; v plq ˘": `upl´1q , v pl´1q ; V pl´1q 0 , V pl´1q 1 ˘, @pV pl´1q 0 , V pl´1q 1 q P T u pl´1q T l´1 Fp Ă M q.
Then, " D plq F t pu 0 , wq ı pu plq ; v plq q " ´"D pl´1q F t pu 0 , wq

ı pu pl´1q , v pl´1q q; " D pl´1q F t pu 0 , wq ‰ pV pl´1q 0 q, ∇ V pl´1q 0 " D pl´1q F t pu 0 , wq ‰ pv pl´1q q `"D pl´1q F t pu 0 , wq ‰ pV pl´1q 1 q (4.19)
and the covariant derivative term

∇ V pl´1q 0 " D pl´1q F t pu 0 , wq ‰ pv pl´1q q involves a combination of the l 1 -th (l 1 ď l ´1) covariant derivatives ∇ V 0,l 1 ∇ V 0,l 1 ´1 ¨¨¨∇ V 0,0 " D p1q F t pu, wq ı pvq, @v, V 0,0 , ¨¨¨, V 0,l 1 P T u Fp Ă M q, (4.20)
where for l 1 " 1, (4.20) was given in (4.13), and for l 1 ą 1, let τ Þ Ñ C l 1 pτ q :" exppτ V 0,l 1 q be the geodesic passing through u and let v pτ q, V 0,0 pτ q, ¨¨¨, V 0,l 1 ´1 pτ q be the parallel transportations of v, V 0,0 , ¨¨¨, V 0,l 1 ´1 along C l 1 to the point C l 1 pτ q, then

∇ V 0,l 1 ∇ V 0,l 1 ´1 ¨¨¨∇ V 0,0 " D p1q F t pu, wq ı pvq " D Bτ ´∇V 0,l 1 ´1 pτ q ¨¨¨∇ V 0,0 pτ q " D p1q F t pC l 1 pτ q, wq ı pv pτ qq ¯ˇˇˇτ "0
.

The Stratonovich SDE of (4.20) involves t∇ ι Hu ιďl 1 , t∇ ι Ru ιďl 1 ´1. But the Itô SDE of (4.20) involves t∇ ι Hu ιďl 1 `1, t∇ ι Ru ιďl 1 .

By Corollary 4.2, all the tangent maps rD plq F t p¨, wqs are invertible. The inverse maps rD plq F t p¨, wqs ´1 can be formulated by the same equation as rD plq F t p¨, wqs, but using the backward infinitesimals d Ý Ñ B τ , ´dτ instead of dB τ , dτ . We skip the details.

4.3.

Growth of the stochastic tangent maps in time. We use the above SDEs to estimate the L q -norm (q ě 1) of sup 0ďtătďT }rD plq F t,t pu, wqs}.

A useful tool to the L q -norm estimations of stochastic integrals is Burkholder's inequality which can be obtained using Itô's formula for | ¨|q and Doob's inequality of martingales.

Lemma 4.7. (cf. [START_REF] Kunita | Stochastic flows and stochastic differential equations[END_REF]Theorem 2.3.12]) For an F τ -adapted R m or OpR m q process f τ ,

(4.21) E ˜ˇˇˇˇż t 1 t xf τ , dB τ y ˇˇˇˇq ¸ď C 1 pqq ¨E ˇˇˇˇż t 1 t |f τ | 2 dτ ˇˇˇˇq 2 , @q ě 2,
where C 1 pqq " p 1 2 qpq ´1qpq{pq ´1qq q´2 q q 2 .

(When q " 2, the inequality in (4.21) becomes an equality and is referred to as the isometry property of Brownian motion.)

We would like to list a simple fact that will be used from time to time for computations in the remaining paper: for any q ě 1 and a 1 , ¨¨¨,

a i 0 P R `Y t0u, i 0 P N, (4.22) p i 0 ÿ i"1 a i q q ď pi 0 q q´1 i 0 ÿ i"1 a q i .
Recall the Dambis-Dubins-Schwarz Theorem which relates local martingales with Brownian motion using Lévy's characterization (see Section 4.4).

Lemma 4.8. (cf. [START_REF] Revuz | Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften[END_REF]Theorems 1.6 &1.7,p. 181]) If M is a pΩ, F, Pq-continuous local martingale vanishing at 0. Let T t " infts : xM, My s ą tu.

i) If xM, My 8 " 8, then B t " M Tt is a pF Tt q-Brownian motion and M t " B xM,Myt . ii) If xM, My 8 ă 8, then there exist an enlargement p r Ω, r F, r Pq of pΩ, F, Pq and a Brownian motion r B on r Ω independent of M such that the process

B t " " M Tt , if t ă xM, My 8 , M 8 `r B t´xM,My8 , if t ě xM, My 8
is a standard linear Brownian motion. The process W given by

W t " " M Tt , if t ă xM, My 8 , M 8 , if t ě xM, My 8
is a p r F t q-Brownian motion stopped at xM, My 8 .

Given an pΩ, F, Pq-Brownian motion B, we know that for almost all w, t Þ Ñ B t pwq is not differentiable, but is a-Hölder continuous for every a P p0, 1{2q. Let ą 0 be fixed. Define

(4.23) }B r0, s pwq} a :" sup 0ďtď |B t pwq| `sup 0ďtăt 1 ď |B t 1 pwq ´Bt pwq| |t 1 ´t| a .
The following result of Skorokhod ([Sk]) is a weaker form of the Fernique Theorem ( [Fe]).

Lemma 4.9. ( [Sk]) Let B be an pΩ, F, Pq-Brownian motion. For any a P p0, 1{2q, there exists ą 0 such that E ´e }B r0, s }a ¯ă 8.

Remark 4.10. The original proof of Lemma 4.9 is for " 1. In general, for any t ą 0 and a ą 0, B t has the same distribution as ? aB t{a . In particular, this holds for a " . A simple calculation shows that }B r0, s } a ď p ? `? { a q}B r0,1s } a . Hence the Lemma is true for every with p q " mint p1q{2, p1q{p ? `? { a qu.

The following estimations are similar to the estimation for the first order tangent map with t " 0, t " T fixed (see [START_REF] Elworthy | Stochastic Differential Equations on Manifolds[END_REF]Proposition 5A,Chapter VIII]).

Proposition 4.11. Let g P M k pM q with k ě 3. For x P Ă M and T P R `, let tu t u tPr0,T s be the solution to (4.7) in O r g p Ă M q with u 0 P O r g x p Ă M q. Then for every l, 1 ď l ď k 2, and q ě 1, there exist c l pqq ą 0, which depends on l, m, q and the norm bounds of t∇ l 1 Hu l 1 ďl`1 , t∇ l 1 Ru l 1 ďl , and c l pqq ą 0, which depends on l, m, q and the norm bounds of t∇ l 1 Hu l 1 ď2 , ∇R, such that

(4.24) E sup 0ďtătďT › › › " D plq F t,t pu t , wq ‰ ˘1› › › q ă c l pqqe c l pqqT .
Proof. By using the cocycle property of the tangent maps, it suffices to show (4.24) with t " 0. We show it by induction. At each step, we only check the bound for the tangent map since the estimation on the inverse map can be obtained analogously using its SDE.

We begin with the l " 1 case. It suffices to consider " Č D p1q F t pu 0 , wq ‰ . Following [Mal2, Theorem 5.1] (see Lemma 4.4), the solutions to (4.11) can be understood using multiplicative stochastic integral in Ito's form. For each j P t1, 2, ¨¨¨, mu and u P Fp Ă M q, define a mpm `1q ˆmpm `1q matrix M j puq, which is an endomorphism from T o FpR m q to itself, such that for pz, zq P T o FpR m q, (4.25)

M j puq `pz, zq ˘" ´pz ι j q m ι"1 , pR ι q,j,l puqz l q m ι,q"1 ¯.

Define another mpm `1q ˆmpm `1q matrix Npuq (or an endomorphism from T o FpR m q to itself) such that for pz, zq P T o FpR m q, Npuq `pz, zq ˘" ´0, N ι q,l puqz l ¯, (4.26) where N ι q,l puq :" m ÿ j"1 @ p∇pue j qR pue j , ue l q ue q , ue ι D .

Using M, N, we conclude from (4.11) that the Itô form of the SDE of zt :" pz t ,

z t q is 1 (4.
|z 0 | 2q ď Cpqqe Cpqqt e r q ş t 0 M j puτ q dB j τ |z 0 | 2q ,
where Cpqq depends on the norm bound of R, ∇R and m, q, the number r q depends on q, and tM j u m j"1 are continuous real valued processes with bounds depending on the norm bound of R. Consider the process M t pwq :"

ż t 0 M j pu τ q dB j τ .
It is a continuous martingale with M 0 " 0 and with the quadratic variation xM, My t ď C 1 t for some constant C 1 which depends on the norm bound of R. By Lemma 4.8, there exist

1 In terms of the multiplicative stochastic integral, (4.27) shows

" Č D p1q F t pu0, wq ‰ " e ! ş t 0 M j puτ q dB j τ pwq`Npuτ q dτ ) .
a continuous martingale r M and a Brownian motion B on an enlargement p r Θ `, r F, r Qq of pΘ `, F, Qq so that r M has the same law as M and

r M t " B x r M, r
Myt . Fix a P p0, 1{2q and consider }B r0,C 1 s } a . Let p1q be as in Lemma 4.9 and put " mint p1q{2, p1q{p ? C 1 `?C 1 {pC 1 q a qu. By Remark 4.10,

E r Q `e › › B r0,C 1 s › › a ˘ă Ă C 1 ă 8, where Ă C 1 depends on C 1 and a. Let t 1 " mintC ´1 1 p r q ´1q 1 a , T u. By the definition of } ¨}a , ˇˇM t ´M0 ˇˇď pC 1 tq a › › B r0,C 1 s › › a ď r q ´1› › B r0,C 1 s › › a .
Using this and (4.28), we obtain

E sup 0ďtďt 1 |z t pwq| 2q ď Cpqqe Cpqqt 1 |z 0 | 2q ¨Er Q `e › › B r0,C 1 s › › a ˘ď Ă C 1 Cpqqe Cpqqt 1 |z 0 | 2q .
This implies

E sup 0ďtďt 1 › › " DF t pu 0 , wq ‰› › 2q ď r Cpqqe Cpqqt 1 ,
where r Cpqq depends on Ă C 1 , Cpqq, m, q. In the same way, we obtain

E sup pi´1qt 1 ďtďit 1 › › " DF pi´1qt 1 ,t pu pi´1qt 1 , wq ‰› › 2q , E sup i 1 pT qt 1 ďtďT › › " DF i 1 pT qt 1 ,t pu i 1 pT qt 1 , wq ‰› › 2q ď r Cpqqe Cpqqt 1 , @i, 1 ď i ď i 1 pT q " maxti P N : it 1 ă T u.
Hence by using the cocycle property and Markov property, we conclude that there are some c 1 pqq, c 1 pqq of the prescribed type in the statement of the proposition such that

E sup 0ďtďT › › " DF 0,t pu 0 , wq ‰› › 2q ă `r Cpqqe Cpqqt 1 ˘i1 pT q`1 ă c 1 pqqe c 1 pqqT .
We proceed to show (4.24) with l " 2 and t " 0. By the above conclusion in the l " 0 case and the definition of

" D p2q F t pu 0 , wq ‰ , it remains to analyze E sup 0ďtďT }pθ, q ut pV t q} 2q , where V t :" ∇ V 0 " D p1q F t pu 0 , wq ‰ pvq and v, V 0 have norm 1. Put v τ :" rD p1q F τ pu 0 , wqsv, V τ :" rD p1q F τ pu 0 , wqsV 0 . Let A τ pwq :" pθ, q uτ ´∇p2q pv τ , V τ qHpu τ , ¨q `RpHpu τ , ¨q, V τ qv τ ¯, (4.29) B τ pwq :" pθ, q uτ ´"Hpu τ , e i q, ∇ p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ı¯, C τ pwq :"
´ ´∇p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ¯ei ,

u ´1 τ R ´uτ e i , u τ θ `∇p2q pv τ , V τ qHpu τ , e i q `RpHpu τ , e i q, V τ qv τ ˘¯u τ ¯ and let r A t pwq :" ż t 0 " Č D p1q F τ,t pu τ , wq ‰ A τ pwq dB τ pwq, r B t pwq :" ż t 0 " Č D p1q F τ,t pu τ , wq ‰ B τ pwq dτ, r C t pwq :" ż t 0 " Č D p1q F τ,t pu τ , wq ‰ C τ pwq dτ.
By Corollary 4.6, pθ, q ut pV t q " r A t pwq `r B t pwq `r C t pwq.

Hence, by using (4.22), we obtain

3 1´2q E sup 0ďtďT }pθ, q ut pV t q} 2q ď E sup 0ďtďT › › › r A t pwq › › › 2q `E sup 0ďtďT › › › r B t pwq › › › 2q `E sup 0ďtďT › › › r C t pwq › › › 2q ": p r Aq `pr Bq `pr Cq.
For p r Aq, it is true by Doob's inequality of sub-martingales and Burkholder's inequality that

p r Aq ďCp2qqE › › › › ż T 0 " Č D p1q F τ,T pu τ , wq ‰ A τ pwq dB τ pwq › › › › 2q ďCp2qqC 1 p2qqE ˇˇˇż T 0 › › › " Č D p1q F τ,T pu τ , wq ‰ A τ pwq › › › 2 dτ ˇˇˇq ďCp2qqC 1 p2qqT q ˜E sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 4q ¸1 2 ¨ˆE sup 0ďτ ďT }A τ pwq} 4q ˙1 2 ,
where Cpqq :" pq{q ´1q q and C 1 pqq is given in Lemma 4.7. Using (4.29), we compute that

E sup 0ďτ ďT }A τ pwq} 4q ďC 4q A ˆE sup 0ďτ ďT }v τ pwq} 8q ˙1 2 ¨ˆE sup 0ďτ ďT }V τ pwq} 8q ˙1 2 ďpC 1 A q 4q E sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 8q ,
where C A , C 1 A depend on the norm bounds of t∇ l Hu lď2 and t∇ l Ru lď1 . With (4.24) for l " 1, we conclude that

p r Aq ď C 1 pqqpC 1 A ? T q 2q a c 1 p4qqc 1 p8qqe 1 2 pc 1 p4qq`c 1 p8qqqT .
Using (4.24) with l " 1 and Hölder's inequality, we have

p r Bq ďT 2q ˜E sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 4q ¸1 2 ¨ˆE sup 0ďτ ďT }B τ pwq} 4q ˙1 2 ďpCT q 2q ˜E sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 4q ¸1 2 ¨˜E sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 8q ¸1 2 ďpCT q 2q a c 1 p4qqc 1 p8qqe 1 2 pc 1 p4qq`c 1 p8qqqT
and the same inequality holds true for pCq, where C depends on the norm bounds of t∇ l Hu lď3 , t∇ l Ru lď2 . Similarly, we can obtain the estimation on " D p2q F t pu 0 , wq ‰ ´1. This finishes the proof of (4.24) for the l " 2 case.

Let l ě 3. Assume (4.24) holds true for tangents up to the pl ´1q-th order. For the estimation on l-th tangent map, by the inductive definition of

" D plq F t pu 0 , wq ‰ (see (4.19)), it remains to show E sup 0ďtďT › › ›∇ p¨q " D pl´1q F t pu 0 , wq ‰ p¨q › › › q ă c l pqqe c l pqqT . (4.30)
This can be done as in the l " 2 case by formulating V pl´1q t in terms of " D pl´1q F τ,t pu τ , wq ‰ by Duhamel's principle and using the inductive assumption on (4.24). 4.4. Brownian bridge and conditional estimations. We want to further estimate the growth of (4.24) with respect to Brownian bridge distributions using their SDEs, which can be derived from the classical Cameron-Martin-Girsanov formula.

We begin with some classical estimations on heat kernels in the non-compact case.

Lemma 4.12. ([Sa, Theorem 6.1]) Let g P M k pM q and let ppt, x, yq be the heat kernel functions of the r g-Brownian motion on Ă M . There exist constants b 1 , c 1 , c 2 , κ 1 (depending on m and the curvature bound) such that for any t ą 0 and x, y P Ă M , we have

(4.31) ppt, x, yq ď 1 b Vol r g Bpx, ? tqVol r g Bpy, ? tq e c 1 p1`b 1 t`?κ 1 tq´p d r g px,yqq 2 c 2 t .
For later use, we would like to state a simplified rough version of (4.31): there are constants c 0 (which depends on }g λ } C 0 ) and c 0 (which depends on }g λ } C 2 ) such that (4.32)

ppT, x, yq ď c 0 T ´me c 0 p1`T q .

Lemma 4.13. ([Li, Theorem 1.5]) Let g P M k pM q and let ppt, x, yq be the heat kernel functions of the r g-Brownian motion on Ă M . Let T ą 0. There are constants cpi, T q, i ď k ´2, which depend on i, T and the curvature and its derivatives up to i-th order, such that, for all pt, x, yq P p0, T s ˆĂ M ˆĂ M , the i-th covariant derivative of ln p satisfies (4.33)

}∇ piq ln ppt, x, yq} ď cpi, T q ˆ1 t d r g px, yq `1 ? t ˙i .

Let T ą 0. For x, y P Ă M , the distribution of the Brownian bridge from x to y in time T , i.e., the Brownian motion starting from x conditioning on paths that are at y at time T , is

P x,y,T p¨q :" E Px `¨ˇˇx T " y ˘.
It is a probability on the bridge space C x,y pr0, T s, Ă M q :" tw P C x pr0, T s, M q : w 0 " x, w T " yu .

Proposition 4.14. Write P x,y,T :" ppT, x, yqP x,y,T . Fix T 0 ą 0. For any q P R `and T ą T 0 , there exists c depending on m, q, T, T 0 and }g} C 2 such that for all x, y P Ă M ,

E P x,y,T e q ş T 0 }∇ ln ppT ´t,xt,yq} dt ďe cp1`dpx,yqq . (4.34)
Proof. By (4.33), there is some c which depends on }g} C 2 and T such that

ż T 0 }∇ ln ppT ´τ, x τ , yq} dτ ď c? T `c ż T 0 1 T ´τ dpx τ , yq dτ.
Hence it is true by Hölder's inequality that for t 0 P p0, mint1, T 0 {2uq small, ´EP x,y,T e q ş T 0 }∇ ln ppT ´t,xt,yq} dt ¯2ďe 2cq

? ":e 2cq ? T pEqpt 0 qpFqpt 0 q, where py t q tPr0,T s denotes the Brownian motion starting from y P Ă M . Let t 0 ă T 0 {2. Then for pEqpt 0 q, by (4.32), we have

T E P x,
pEqpt 0 q " E P ẙ,x,T e 2cq ş t 0 0 1 τ dpyτ ,yq dτ " E Py ´e2cq ş t 0 0 1 τ dpyτ ,yq dτ ¨ppT ´t0 , y t 0 , xq ď c 0 2 m T ´m 0 e c 0 p1`T q E Py e 2cq ş t 0 0 1 τ dpyτ ,yq dτ .
To show there is some small t 0 ą 0 (depending on m, q, T, T 0 and }g} C 2 ) such that pEqpt 0 q is bounded, we can use a trick from Driver ([D2, Lemma 3.8]) to compare it with Euclidean Brownian motions. Find finite many smooth functions tf i u l i"1 on Ă M with f i pyq " 0 and dpz, yq ď ř l i"1 |f i pzq| for all z P Ă M , where all f i have bounded first and second order differentials on Ă M . So for an upper bound estimation of pEqpt 0 q, it suffices to consider

E Py e 2cq ş t 0 0 1 τ |f pyτ q| dτ ": pE 1 qpt 0 q
for any C 2 function f on Ă M with bounded differentials up to second order. Let py, , Bq be the triple which defines the Brownian motion on Ă M starting from y. By Itô's formula,

|f py t q| ď ˇˇˇż t 0 ´1 τ ∇f py τ q dB τ `ż t 0 ∆f py τ q dτ ˇˇˇď ˇˇˇż t 0 ´1 τ ∇f py τ q dB τ ˇˇˇ`C pf qt,
where Cpf q is some constant which bounds |∆f |. Hence,

pE 1 qpt 0 q ď e 2cqt 0 Cpf q ¨EPx `e2cq ş t 0 0 1 τ }M 1 τ } dτ ˘, where M 1 t :" ż t 0 ´1 τ ∇f py τ q dB τ .
The process M 1 t is a continuous martingale with M 1 0 being the zero vector and has quadratic variation xM 1 , M 1 y t ď C 1 t for some constant C 1 which depends on the bound of |∇f |. So, to show pE 1 qpt 0 q is finite for small t 0 , it suffices to show E Px `e2cq

ş t 0 0 1 τ }M 1 τ } dτ ˘is for M 1
t being in the one-dimensional process case. By Lemma 4.8, there exist a continuous martingale r M 1 and a Brownian motion B 1 on an enlargement p r Ω 1 , r F 1 , r P 1 q of pΩ, F, Pq such that r M 1 has the same law as M 1 and

r M 1 t " B 1 x r M 1 , r
M 1 yt . Let a P p0, 1{2q. By Lemma 4.9, there is some 1 ą 0 which depends on }g} C 2 such that

E r P 1 `e 1 › › B 1 r0, 1 2 C 1 s › › a ˘is finite.
By the definition of the Hölder norm } ¨}a ,

ż t 0 0 1 τ }M 1 τ } dτ ď › › B 1 r0, 1 2 C 1 s › › a ¨pC 1 q a ż t 0 0 τ a´1 dτ ď 1 a pC 1 t 0 q a › › B 1 r0, 1 2 C 1 s › › a .
Hence, for t 0 " mint1, T 0 {2, pap2cqq ´1 1 q 1 a {C 1 u, we have

E Px `e2cq ş t 0 0 1 τ }M 1 τ } dτ ˘ď E r P 1 `e 1 › › B 1 r0, 1 2 C 1 s › › a ˘ă 8.
For pFqpt 0 q, by symmetry of the bridge distribution,

pFqpt 0 q " E P x,y,T e 2cq ş 1 2 T 0 1 t 0 dpxτ ,yq dτ `2cq ş T ´t0 1 2 T 1 t 0 dpy T ´τ ,yq dτ ď e cqt ´1 0 T dpx,yq ˜EP x,y,T e 4cq ş 1 2 T 0 1 t 0 dpxτ ,xq dτ ¸1 2 ˜EP ẙ,x,T e 4cq ş 1 2 T 0 1 t 0 dpyτ ,yq dτ ¸1 2 .
By (4.31) and Markov property of p (see (4.36)),

E P ẙ,x,T e 4cq ş 1 2 T 0 1 t 0 dpyτ ,yq dτ " E Py ˆe4cqt ´1 0 ş 1 2 T 0 dpyτ ,yq dτ ¨pp 1 2 T , y 1 2 T , xq ď c 0 2 m T ´m 0 e c 0 p1`T q E Py e 4cqt ´1 0 ş 1 2 T 0 dpyτ ,yq dτ .
Let t 1 0 ă mint1, T 0 {2u be small. Partition r0, T {2s into 0 " τ 0 ă τ 1 ă ¨¨¨ă τ N ă T {2, where τ i :" it 1 0 and N :" maxti, τ i ă T {2u, and chop the integral ş 1 2 T 0 dpy τ , yq dτ into pieces accordingly. Using the triangle inequality, we obtain

ż 1 2 T 0 dpy τ , yq dτ ď N ÿ i"1 ˜ż τ i τ i´1
dpy t , y τ i´1 q dt `t1 0 dpy τ i´1 , yq

¸`ż 1 2 T τ N dpy t , y τ N q dt `t1 0 dpy τ N , yq ď N ÿ i"1 ˜ż τ i τ i´1 dpy t , y τ i´1 q dt `pN `1 ´iqt 1 0 dpy τ i , y τ i´1 q ¸`ż 1 2 T τ N dpy t , y τ N q dt.
Using the Markov property of the Brownian motion and Hölder's inequality, we see that

ˆEPy e 4cqt ´1 0 ş 1 2 T 0
dpyτ ,yq dτ ˙2 ď ˜sup

y 1 P Ă M E P y 1 e q ş t 1 0 0 dpy 1 t ,y 1 q dt ¸N`1 ¨N ź i"1 sup y 1 P Ă M E P y 1 e qt 1 0 pN `1´iqdpy 1 t 1 0 ,y 1 q ": `pFq 0 ˘N`1 N ź i"1 pFq i ,
where q :" 8cqt ´1 0 , y 1 t is the Brownian motion on Ă M which starts from y 1 and P y 1 is its distribution probability. For pFq 0 , we estimate as in the first part. Note that p Ă M , r gq is the universal cover of the compact space pM, gq, although the choice of the f i may differ from point to point, we can ensure their differentials up to second order are uniformly bounded. So we can choose t 1 0 (for instance, t 1 0 " mint1, T 0 , 1 {qu) such that pFq 0 is bounded. Fix such a t 1 0 and estimate pFq i using Lemma 4.12. We obtain some constant cpt 1 0 q such that pFq i ď cpt 1 0 qe c 2 q2 pt 1 0 q 3 pN `1´iq 2 , where c 2 is as in (4.31). Hence, there is some constant cpqq depending on m, q, T, T 0 and }g} C 2 such that

E Py `e4cq 1 t ´1 0 ş 1 2 T 0 dpyτ ,yq dτ ˘ď e cpqq .
So, pFqpt 0 q ď c 0 2 m T ´m 0 e cpqq`c 0 p1`T q`cqt ´1 0 T dpx,yq . Putting the estimations on pEqpt 0 q, pFqpt 0 q together, we obtain (4.34).

Consider the Wiener space C 0 pr0, T s, R m q with the standard filtration pF t q tPr0,T s and let pB t q tPr0,T s be an pF t q-Brownian motion starting from 0 with respect to a probability measure Q on F T . Let f : r0, T s Þ Ñ R m be square integrable with respect to Lebesgue measure. Define a random process pM t q tPr0,T s on r0, T s satisfying M 0 " 1 and Itô's SDE

dM t " 1 2 M t xf t , dB t y. Then M t " e t 1 2 ş t 0 xfτ ,dBτ pwqy´1 4 ş t 0 |fτ | 2 dτ u . Since E Q pe 1 4 ş t 0 |fτ | 2 dτ
q, t ď T , are all finite, we have by Novikov ([No]), that pM t q tPr0,T s is a continuous pF t q tPr0,T s -martingale, i.e., E Q pM t q " 1, @t P r0, T s.

For t P r0, T s, let r Q t be the probability on C 0 pr0, T s, R m q, which is absolutely continuous with respect to Q with d r Q t dQ pwq " M t pwq.

Since M t is a martingale, the projection of r Q t on F τ , τ ă t, is given by the same formula. The classical [START_REF] Cameron | Transformations of Wiener integrals under a general class of linear transformations[END_REF][START_REF] Girsanov | On transforming a class of stochastic processes by absolutely continuous substitution of measures[END_REF]) says that the process pB t ´şt 0 f τ dτ q tPr0,T s is a Brownian motion with respect to r Q T . In other words, we have that the probability Q on Wiener space is quasi-invariant under the transformation T :

C 0 pr0, T s, R m q Ñ C 0 pr0, T s, R m q : w Þ Ñ w `ş¨0 f τ dτ with (4.35) dQ ˝T´1 dQ pwq " e t 1 2 ş T 0 xfτ pwq, dBτ pwqy´1 4 ş T 0 |fτ pwq| 2 dτ u .
As in the compact case (see [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]Theorem 5.4.4]), we can deduce the SDE of the Brownian bridge on Ă M from the Cameron-Martin-Girsanov Theorem. Let px t , u t q tPr0,T s be the stochastic pair which defines the Brownian motion starting from x up to time One can conclude that tU t u tPr0,T s is a semi-martingale on r0, T s since

E P y,x,T ˆż T 0 }∇ ln ppT ´τ, x τ , yq}dτ ˙ă 8
is also true on the non-compact universal cover space p Ă M , r gq by (4.34). In summary, Lemma 4.15. There is a Brownian motion pb t q tPr0,T q such that the horizontal lift U of the Brownian bridge x is a semi-martingale on r0, T s which satisfies the SDE (4.37) dU t " HpU t , e i q ˝`db i t `2HpU t , e i q ln r p M pT ´t, U t , xqdt ˘.

In other words, the anti-development of the Brownian bridge x (i.e., the pre-image via parallelism, see Section 5.2 for more precise definition) is

W t " b t `2 ż t 0 U ´1 τ ∇ ln ppT ´τ, x τ , yq dτ.
Now, we can use Proposition 4.14 and Lemma 4.15 to derive a bridge version of (4.24).

Proposition 4.16. Let g P M k pM q, k ě 3. For x P Ă M , let pu t q tPr0,T s be the solution to (4.7) in O r g p Ă M q with u 0 P O r g x p Ă M q. For every T 0 ą 0, l, 1 ď l ď k ´2, q ě 1 and T ą T 0 , there exist c 1 l pqq ą 0, which depends on l, m, q and the norm bounds of t∇ l 1 Hu l 1 ďl , t∇ l 1 Ru l 1 ďl , and c 1 l pqq ą 0, which depends on l, m, q, T, T 0 and the norm bounds of t∇ l 1 Ru l 1 ď1 , such that

(4.38) E P x,y,T sup 0ďtătďT › › › " D plq F t,t pu t , wq ‰ ˘1› › › q ă c 1 l pqqe c 1 l pqqp1`dpx,yqq , @x, y P Ă M .
Proof. Using the cocycle property of the tangent map, it suffices to show (4.38) for t " 0. We show this by induction and in each step, we only verify it for the forward tangent map.

When l " 1, it suffices to consider " Č D p1q F t p¨, wq ‰ , whose SDE is as in (4.27) with

dB τ " db τ `2u ´1 τ ∇ ln ppT ´τ, x τ , yq dτ, (4.39)
where pb τ q τ Pr0,T q is a Brownian motion for P x,y,T by Lemma 4.15. Hence the conditional norm of › › " Č D p1q F t pu 0 , wq ‰› › q differs in distribution with the nonconditional case by a multiple e r cpqq ş T 0 }∇ ln ppT ´t,xt,yq} dt for some constant r cpqq which depends on the norm bound of R, ∇R and m, q. Hence by Hölder's inequality and Proposition 4.11, we have

˜EP x,y,T sup 0ďtďT › › › " Č D p1q F t pu 0 , wq ‰ › › › q ¸2 ď c l p2qqe c l p2qqT E P ẙ,x,T e 2r cpqq ş T 0 }∇ ln ppT ´t,xt,yq} dt .
This shows (4.38) for the l " 1 case by Proposition 4.14.

Using the decomposition of rD p2q F t s and the first step conclusion, the l " 2 case of (4.38) can be reduced to the estimation of pVq :" E P x,y,T sup 0ďtďT }pθ, q ut pV t q} 2q .

Let A, B, C and r A, r B, r C be given in the proof of Proposition 4.11. Then

3 1´2q pVq ď E P x,y,T sup 0ďtďT › › › r A t pwq › › › 2q `EP x,y,T sup 0ďtďT › › › r B t pwq › › › 2q `EP x,y,T sup 0ďtďT › › › r C t pwq › › › 2q ": pAq `pBq `pCq.
For pBq, following its non-conditional estimation in the proof of Proposition 4.11, we obtain

pBq ďpCT q 2q ˜EP x,y,T sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 4q E P x,y,T sup 0ďtătďT › › › " Č D p1q F t,t pu t , wq ‰ › › › 8q ¸1 2 ďpCT q 2q b c 1 1 p4qqc 1 1 p8qqe 1 2 pc 1 1 p4qq`c 1 1 p8qqqp1`dpx,yqq ,
where C depends on the norm bounds of t∇ l Hu lď3 , t∇ l Ru lď2 and c 1 1 , c 1 1 are from (4.38) for l " 1, and the constants can be chosen such that the same bound is valid for pCq. For pAq, we use (4.39). Let A 1 t pwq :"

ż t 0 " Č D p1q F τ,t pu τ , wq ‰ A τ pwq db τ pwq, A 2 t pwq :" ż t 0 " Č D p1q F τ,t pu τ , wq ‰ A τ pwq 2ptu τ s λ q ´1∇ λ ln ppT ´τ, tx τ s λ , yq dτ.
Then,

2 1´2q pAq ď E P x,y,T sup 0ďtďT › › ›A 1 t pwq › › › 2q `EP x,y,T sup 0ďtďT › › ›A 2 t pwq › › › 2q ": pAq 1 `pAq 2 .
Using the Brownian character of b τ with respect to P x,y,T , we can estimate pAq 1 as in the non-conditional case using Doob's inequality of sub-martingales and Burkholder's inequality. This gives

pAq 1 ď Cp2qqC 1 p2qqT q ˆEP x,y,T sup 0ďτ ătďT › › › " Č D p1q F τ,t pu τ , wq ‰ › › › 4q ˙1 2 ¨ˆE P x,y,T sup 0ďτ ďT }A τ } 4q ˙1 2 ,
where C, C 1 are as in the proof of Proposition 4.11. Using (4.29), we compute that

E P x,y,T sup 0ďτ ďT }A τ } jq ďpC 1 A q jq E P x,y,T sup 0ďτ ďT › › › " Č D p1q F 0,τ pu 0 , wq ‰ › › › 2jq
, @j P N, (4.40)

where C 1 A depends on the norm bounds of t∇ l Hu lď2 , t∇ l Ru lď1 . Hence, by (4.38) for l " 1,

pAq 1 ďC 1 pqqpC 1 A ? T q 2q b c 1 1 p4qqc 1 1 p8qqe 1 2 pc 1 1 p4qq`c 1 1 p8qqqp1`d r g px,yqq .
For pAq 2 , we have

`pAq 2 ˘3 ďE P x,y,T sup 0ďτ ătďT › › › " Č D p1q F τ,t pu τ , wq ‰ › › › 6q ¨EP x,y,T sup 0ďτ ďT }A τ } 6q ¨EP x,y,T ˇˇˇż T 0 }∇ ln ppT ´τ, x τ , yq} dτ ˇˇˇ6 q .
Note that

E P x,y,T ˇˇˇż T 0 }∇ ln ppT ´τ, x τ , yq} dτ ˇˇˇ6 q ď E P x,y,T e 6q ş T 0 }∇ ln ppT ´t,xt,yq} dt .
So by Proposition 4.14, (4.40) and (4.38) for l " 1, we compute that

pAq 2 ď pC 1 A q 2q 3 b c 1 1 p6qqc 1 1 p12qqcp6qqe 1 3 pc 1 1 p6qq`c 1 1 p12qq`cp6qqqp1`dpx,yqq .
Hence pVq has the same type of bound as in (4.38) for l " 2 as claimed.

Assume we have shown (4.38) for l ď l 0 ´1 ď k ´3. Using the induction assumption and (4.19), we can reduce the estimation of (4.24) at l " l 0 to the conditional estimation of (4.30), which can be done exactly as in the l " 2 step. 4.5. Regularity of the stochastic analogue of the geodesic flow. Finally, we employ the SDE theory in the previous subsections of this section to discuss the regularity of the Brownian companion process u with respect to metric changes.

Let λ P p´1, 1q Þ Ñ g λ P M k pM q be a C k curve. Each lifted metric r g λ in Ă M determines a horizontal space H λ T Fp Ă
M q of the frame bundle space. For any u P Fp Ă M q, let H λ pu, e i q, i " 1, ¨¨¨, m, be the vector in H λ u T Fp Ă M q which projects to ue i . Since

g λ P M k pM q, the map u Þ Ñ H λ pu, e i q, u P Fp Ă M q, is C k´1 bounded. Hence the SDE (4.41) dtu t s λ " m ÿ i"1 H λ ptu t s λ , e i q ˝dB i t pwq is solvable in Fp Ă M q for any initial tu 0 s λ P F x p Ă M q, x P Ă M . In particular, if tu 0 s λ P O r g λ x p Ă M q, tu t s λ remains in O r g λ p Ă
M q and its projection to Ă M gives the stochastic process of the r g λ -Brownian motion starting from x. Let tF t s λ : tu 0 s λ Þ Ñ tu t s λ denote the flow map associated to (4.41). Let rD plq tF t s λ p¨, wqs, 1 ď l ď k ´2, be the l-th tangent map of tF t s λ and denote by r Č D plq tF t s λ p¨, wqs its pull back map in T l FpR m q via the map pθ, q. They have the following regularity in λ by applying Proposition 4.1.

Lemma 4.17. Let λ P p´1, 1q Þ Ñ g λ P M k pM q pk ě 3q be a C k curve. Assume H λ p¨, e i q has bounded norms (independent of λ) for the covariant derivatives up to the pk ´1q-th order with respect to the reference metric r g 0 .

i) Let λ Þ Ñ tu 0 s λ be a C k´2 curve in Fp Ă M q and let ttu t s λ u tPR `be the solution to (4.41) with initial value tu 0 s λ . Then there is a version of the solution to (4.41) such that almost surely, tu t s λ pwq is C k´2 in λ for any t P R `. ii) For each l, 1 ď l ď k ´2, the tangent map rD plq tF t s λ p¨, wqs is C k´2´l in λ. In particular, for any

v P T l Fp Ă M q, the map λ Þ Ñ " D plq tF t s λ p¨, wq ‰ v is C k´2´l .
Proof. Consider the stochastic process tr u t u tPR `on Fp Ă M q ˆp´1, 1q with (4.42)

dr u t " m ÿ i"1
r H i pr u t q ˝dB i t pwq, where r H i " pH λ p¨, e i q, 0q.

It has the solution r u t " ptu t s λ , λq for r u 0 " ptu 0 s λ , λq, where tu t s λ is the solution of (4.41) with initial value tu 0 s λ . Since (4.42) is a C k´1 SDS on Fp Ă M q ˆp´1, 1q, we have by Proposition 4.1 ii) that for almost all w, the mapping r u 0 pwq Ñ r u t pwq is C k´2 . Consequently, for any C k´2 curve λ Þ Ñ tu 0 s λ , tu t s λ pwq is C k´2 in λ for almost all w.

For each l, 1 ď l ď k ´2, the SDE of rD plq tF t s λ p¨, wqs was given in Section 4.1 and it forms a C k´1´l SDS on T Fp Ă M q. As in Lemma 4.17, we can treat the one parameter family SDEs of rD plq tF t s λ p¨, wqs as a C k´1´l SDS on T Fp Ă M q ˆp´1, 1q when λ Þ Ñ g λ is C k in M k pM q. So Proposition 4.1 applies and shows ii).

For 0 ď t ă t ă 8, let tF t,t s λ : tu t s λ Þ Ñ tu t s λ denote the flow map associated to (4.41) and let rD plq tF t,t s λ p¨, wqs, l ď k ´2, be its l-th tangent map. As a corollary of the cocycle property of tF t,t s λ and Lemma 4.17, " D plq tF t,t s λ p¨, wq ‰ is C k´2´l differentiable in λ and we denote its j-th differential by `rD plq tF t,t s λ p¨, wqs ˘pjq λ for j ď k ´2 ´l. Let tu t s λ be as in Lemma 4.17 and let ptu t s λ q pjq λ , j ď k ´2, be its j-th differential in λ. We identify ptu t s λ q pjq λ " `rD p0q tF t,t s λ ptu t s λ , wqs ˘pjq λ .

In the following, we show the L q -norm bounds in Propositions 4.11 and 4.16 are also valid for `rD plq tF t,t s λ p¨, wqs ˘pjq λ by a detailed analysis of their SDEs. Endow Fp Ă M q ˆp´1, 1q with the product metric d M q and d p´1,1q is canonical. Let ∇ be the r g 0 Levi-Civita connection and θ, be the associated canonical form and curvature form. Let pH λ q pjq λ pu, ¨q, j ď k ´2, be the j-th differential in λ of the maps H λ pu, ¨q. The SDEs of `rD plq tF t,t s λ p¨, wqs ˘pjq λ can be formulated by using Proposition 4.3. We state them as follows.

Lemma 4.18. Let tu t s λ be as in Lemma 4.17.

i) The Stratonovich SDE of ptu t s λ q p1q λ in T Fp Ă M q is dptu t s λ q p1q λ " ∇pptu t s λ q p1q λ qH λ ptu t s λ , ˝dB t q `pH λ q p1q λ ptu t s λ , ˝dB t q.
ii) The Stratonovich SDE of pθ, q tuts λ `ptu t s λ q

p1q λ ˘in T FpR m q is d `θptu t s λ q p1q λ ˘" dθ ´Hλ ptu t s λ , ˝dB t q, ptu t s λ q p1q λ ¯, d ` ptu t s λ q p1q λ ˘" d ´Hλ ptu t s λ , ˝dB t q, ptu t s λ q p1q λ ¯`∇pptu t s λ q p1q λ q ` `Hλ ptu t s λ , ˝dB t q ˘˘ ´pH λ q p1q λ ptu t s λ , ˝dB t q ¯.
iii) The Itô SDE of pθ, q tuts λ `ptu t s λ q p1q λ ˘in T FpR m q is d `θptu t s λ q p1q λ ˘" dθ ´Hλ ptu t s λ , dB t q, ptu t s λ q p1q λ ∇pH λ ptu t s λ , e i qqdθ ˘´H λ ptu t s λ , e i q, ptu t s λ q p1q λ ¯dt `dθ ´∇pH λ ptu t s λ , e i qqH λ ptu t s λ , e i q, ptu t s λ q p1q λ ¯dt `dθ ´Hλ ptu t s λ , e i q, ∇pptu t s λ q p1q λ qH λ ptu t s λ , e i q `pH λ q p1q λ ptu t s λ , e i q ¯dt, d ` ptu t s λ q p1q λ ˘" d ´Hλ ptu t s λ , dB t q, ptu t s λ q p1q λ ∇pH λ ptu t s λ , e i qqd ˘´H λ ptu t s λ , e i q, ptu t s λ q p1q λ ¯dt `d ´∇pH λ ptu t s λ , e i qqH λ ptu t s λ , e i q, ptu t s λ q p1q λ ¯dt `d ´Hλ ptu t s λ , e i q, ∇pptu t s λ q p1q λ qH λ ptu t s λ , e i q `pH λ q p1q λ ptu t s λ , e i q ¯dt `∇pptu t s λ q p1q λ q ` `Hλ ptu t s λ , dB t q ˘˘` ´pH λ q p1q λ ptu t s λ , dB t q ∇pH λ ptu t s λ , e i qq ´∇pptu t s λ q p1q λ q ` `Hλ ptu t s λ , e i q ˘˘¯d t `∇pH λ ptu t s λ , e i qq

´ ´pH λ q p1q λ ptu t s λ , e i q ¯¯dt.

Note that Kerpθq " V T Fp Ă M q, Kerp q " HT Fp Ă M q and for any v 1 , v 2 P HT Fp Ă M q, v 3 P V T Fp Ă M q, the bracket r¨, ¨s satisfies the property (cf. [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]Lemma 5.5.1])

rv 1 , v 2 s P V T Fp Ă M q, rv 1 , v 3 s P HT Fp Ă M q.
Using these facts, we can simplify the SDEs of pθ, q `ptu t s λ q p1q λ ˘at λ " 0. Corollary 4.19. Let tu t s λ be as in Lemma 4.17.

i) The Stratonovich SDE of pθ, q tuts 0 `ptu t s λ q p1q 0 ˘on T FpR m q is d `θptu t s λ q p1q 0 ˘" ptu t s λ q p1q 0 q ˝dB t , d ` ptu t s λ q p1q 0 ˘" ptu t s 0 q ´1R `tu t s 0 ˝dB t , θptu t s λ q p1q 0 q ˘tu t s 0 ` `pH λ q p1q 0 ptu t s 0 , ˝dB t q ˘.

ii) The Itô SDE of pθ, q tuts 0 `ptu t s λ q p1q 0 ˘on T FpR m q is d `θptu t s λ q p1q 0 ˘" ptu t s λ q p1q 0 q dB t `Ricpu t θptu t s λ q p1q 0 q dt ` `pH λ q p1q 0 ptu t s 0 , e i q ˘ei dt, d ` ptu t s λ q p1q 0 ˘" ptu t s 0 q ´1R `tu t s 0 dB t , tu t s 0 θptu t s λ q p1q 0 q ˘tu t s 0 ` `pH λ q p1q 0 ptu t s 0 , dB t q ptu t s 0 q ´1R ´tu t s 0 e i , tu t s 0 ptu t s λ q p1q 0 qe i ¯tu t s 0 dt `ptu t s 0 q ´1`∇ ptu t s 0 e i qR ˘´tu t s 0 e i , tu t s 0 θptu t s λ q p1q 0 q ¯tu t s 0 dt.

Using Corollary 4.19 and Itô's formula, we can express ptu t s λ q p1q 0 using D p1q tF τ,t s 0 p¨, wq by Duhamel's principle. This can be verified as in Corollary 4.6. We omit the proof.

Corollary 4.20. Let tu t s λ be as in Lemma 4.17.

i) On T Fp Ă M q, ptu t s λ q p1q 0 " " D p1q tF t s 0 ptu 0 s 0 , wq ı ptu 0 s λ q p1q 0 `Vc pptu t s λ q p1q 0 q, where V c pptu t s λ q p1q 0 q :" ż t 0 " D p1q tF τ,t s 0 ptu τ s 0 , wq ı pH λ q p1q 0 ptu τ s 0 , ˝dB τ pwqq.

ii) On T FpR m q, the Itô form of Č ptu t s λ q p1q 0 :" pθ, q tuts 0 ´ptu t s λ q p1q 0 ¯is given by

Č ptu t s λ q p1q 0 " " Č D p1q tF t s 0 ptu 0 s λ , wq ı Č ptu 0 s λ q p1q 0 `Ă V c pptu t s λ q p1q 0 q, where Ă V c pptu t s λ q p1q 0 q :" ż t 0 " Č D p1q tF τ,t s 0 ptu τ s 0 , wq ı ´
`pH λ q p1q 0 ptu τ s 0 , e i q ˘ei dτ, `pH λ q p1q 0 ptu τ s 0 , dB τ q ˘¯.

To describe the second order differential of tu t s λ in λ, we use the horizontal/vertical sum decomposition of T T Fp Ă M q of r g 0 . By Lemma 4.18, it remains to find the SDEs of

ptu t s λ q p2q λ :" D dλ ´ptu t s λ q p1q λ ¯" ∇pptu t s λ q p1q 0 qptu t s λ q p1q 0 .
Lemma 4.21. Let λ P p´1, 1q Þ Ñ g λ P M k pM q be a C k curve with k ě 4. Let λ Þ Ñ tu 0 s λ be C k´2 and let tu t s λ be as in Lemma 4.17 with ptu t s λ q p1q λ , ptu t s λ q p2q λ defined as above.

i) The Stratonovich SDE of ptu t s λ q

p2q λ on T Fp Ă M q is d `ptu t s λ q p2q λ ˘"∇ `ptu t s λ q p2q λ ˘Hλ ptu t s λ , ˝dB t q `∇p2q `ptu t s λ q p1q λ , ptu t s λ q p1q λ ˘Hλ ptu t s λ , ˝dB t q `R`H λ ptu t s λ , ˝dB t q, ptu t s λ q p1q λ ˘ptu t s λ q p1q λ `2∇ `ptu t s λ q p1q λ ˘`H λ ˘p1q λ ptu t s λ , ˝dB t q ``H λ ˘p2q λ ptu t s λ , ˝dB t q. ii) The Stratonovich SDE of pθ, q tuts λ `ptu t s λ q p2q λ ˘on T FpR m q is d ´`θ, ˘`ptu t s λ q p2q λ ˘" d `θ, ˘´H λ ptu t s λ , ˝dB t q, ptu t s λ q p2q λ ¯`∇pptu t s λ q p2q λ q ``θ, ˘`H λ ptu t s λ , ˝dB t q ˘θ, ˘´∇ p2q `ptu t s λ q p1q λ , ptu t s λ q p1q λ ˘Hλ ptu t s λ , ˝dB t q θ, ˘´R `Hλ ptu t s λ , ˝dB t q, ptu t s λ q p1q λ ˘ptu t s λ q p1q λ θ, ˘´2∇ `ptu t s λ q p1q λ ˘`H λ ˘p1q λ ptu t s λ , ˝dB t q ``H λ ˘p2q λ ptu t s λ , ˝dB t q ¯.
iii) The Itô SDE of pθ, q tuts λ `ptu t s λ q

p2q λ ˘on T FpR m q is d ´`θ, ˘`ptu t s λ q p2q λ ˘" d `θ, ˘´H λ ptu t s λ , dB t q, ptu t s λ q p2q λ ¯`∇pptu t s λ q p2q λ q ``θ, ˘`H λ ptu t s λ , dB t q ˘θ, ˘´∇ p2q `ptu t s λ q p1q λ , ptu t s λ q p1q λ ˘Hλ ptu t s λ , dB t q θ, ˘´R `Hλ ptu t s λ , dB t q, ptu t s λ q p1q λ ˘ptu t s λ q p1q λ θ, ˘´2∇ `ptu t s λ q p1q λ ˘`H λ ˘p1q λ ptu t s λ , dB t q ``H λ ˘p2q λ ptu t s λ , dB t q ∇pH λ ptu t s λ , e i qq ! d `θ, ˘´H λ ptu t s λ , e i q, ptu t s λ q p2q λ ∇pptu t s λ q p2q λ q ``θ, ˘`H λ ptu t s λ , e i q ˘θ, ˘´∇ p2q `ptu t s λ q p1q λ , ptu t s λ q p1q λ ˘Hλ ptu t s λ , e i q θ, ˘´R `Hλ ptu t s λ , e i q, ptu t s λ q p1q λ ˘ptu t s λ q p1q λ θ, ˘´2∇ `ptu t s λ q p1q λ ˘`H λ ˘p1q λ ptu t s λ , e i q ``H λ ˘p2q λ ptu t s λ , e i q ¯) dt.
Again, we can simplify the SDEs in Lemma 4.21 at λ " 0.

Corollary 4.22. We retain all the notations in Lemma 4.21.

i) The Stratonovich SDE of pθ, q tuts 0 `ptu t s λ q

p2q 0 ˘on T FpR m q is d ´`θ, ˘`ptu t s λ q p2q 0 ˘¯"
´ ptu t s λ q p2q 0 q ˝dB t , ptu t s 0 q ´1R `tu t s 0 ˝dB t , θptu t s λ q p2q 0 q ˘tu t s 0 θ, ˘´∇ p2q `ptu t s λ q p1q 0 , ptu t s λ q p1q 0 ˘H0 ptu t s 0 , ˝dB t q θ, ˘´R `H0 ptu t s 0 , ˝dB t q, ptu t s λ q p1q 0 ˘ptu t s λ q p1q 0 θ, ˘´2∇ `ptu t s λ q p1q 0 ˘`H λ ˘p1q 0 ptu t s 0 , ˝dB t q ``H λ ˘p2q 0 ptu t s 0 , ˝dB t q ¯.

ii) The Itô SDE of θ tuts 0 `ptu t s λ q p2q 0 ˘on T FpR m q is d `θpptu t s λ q p2q 0 qq " pptu t s λ q p2q 0 qdB t `Ricptu t s 0 θpptu t s λ q p2q 0 qqdt `Φθ pptu t s λ q p1q 0 , ptu t s λ q p1q 0 , dB t , dtq `Φ0,2 θ pptu t s λ q p1q 0 , dB t , dtq, where Φ θ p¨, ¨, dB t , dtq is given in (4.15) for tu t s 0 and2 Φ 0,2 θ pptu t s λ q p1q 0 , dB t , dtq :" ´2∇ `ptu t s λ q p1q 0 ˘`H λ ˘p1q 0 ptu t s 0 , e i q ``H λ ˘p2q 0 ptu t s 0 , e i q ¯ei dt `θ ´"Hptu t s 0 , e i q, 2∇ `ptu t s λ q p1q 0 ˘`H λ ˘p1q 0 ptu t s 0 , e i q ``H λ ˘p2q 0 ptu t s 0 , e i q ı¯d t,

˘p2q λ .
and the Itô SDE of tuts 0 `ptu t s λ q p2q 0 ˘on T FpR m q is d ` pptu t s λ q p2q 0 qq " ptu t s 0 q ´1R ´tu t s 0 dB t , tu t s 0 θpptu t s λ q p2q 0 q ¯tu t s 0 `ptu t s 0 q ´1R ´tu t s 0 e i , tu t s 0 pptu t s λ q p2q 0 qe i ¯tu t s 0 dt `ptu t s 0 q ´1`∇ ptu t s 0 e i qR ˘´tu t s 0 e i , tu t s 0 θpptu t s λ q p2q 0 q ¯tu t s 0 dt `Φ pptu t s λ q p1q 0 , ptu t s λ q p1q 0 , dB t , dtq `Φ0,2 pptu t s λ q p1q 0 , dB t , dtq,

where Φ θ p¨, ¨, dB t , dtq is given in (4.15) for tu t s 0 and Φ 0,2 pptu t s λ q p1q 0 , dB t , dtq :" ´2∇ `ptu t s λ q p1q 0 ˘`H λ ˘p1q 0 ptu t s 0 , dB t q ``H λ ˘p2q 0 ptu t s 0 , dB t q ¯.

By Corollary 4.22 with Lemma 4.5, we can formulate ptu t s λ q p2q 0 and pθ, qptu t s λ q p2q 0 using D p1q tF τ,t s 0 p¨, wq by stochastic Duhamel principle. We only state the conclusion.

Corollary 4.23. We retain all the notations in Lemma 4.21.

i) On T Fp Ă M q, ptu t s λ q p2q 0 " " D p1q tF t s 0 ptu 0 s 0 , wq ı ptu 0 s λ q p2q 0 `∇ptu 0 s λ q p1q 0 " D p1q tF t s 0 ptu 0 s 0 , wq ı `ptu 0 s λ q p1q 0 Vc `ptu t s λ q p2q 0 ˘,
where V c `ptu t s λ q p2q 0 ˘" ż t 0 " D p1q tF τ,t s 0 ptu τ s 0 , wq ı ! ∇ p2q `ptu τ s λ q p1q 0 , ptu τ s λ q p1q 0 ˘H0 ptu τ s 0 , ˝dB τ q ´∇p2q `rD p1q tF τ s 0 sptu 0 s λ q p1q 0 , rD p1q tF τ s 0 sptu 0 s λ q p1q 0 ˘H0 ptu τ s 0 , ˝dB τ q `R`H 0 ptu τ s 0 , ˝dB τ q, ptu τ s λ q p1q 0 ˘ptu τ s λ q p1q 0 ´R`H 0 ptu τ s 0 , ˝dB τ q, rD p1q tF τ s 0 sptu 0 s λ q p1q 0 ˘rD p1q tF τ s 0 sptu 0 s λ q

p1q 0 `2∇ `ptu τ s λ q p1q 0 ˘`H λ ˘p1q 0 ptu τ s 0 , ˝dB t q ``H λ ˘p2q 0 ptu τ s 0 , ˝dB τ q ) .
ii) On T FpR m q, the Itô form of Č ptu t s λ q p2q 0 :" pθ, q tuts 0 `ptu t s λ q p2q 0 ˘is

Č ptu t s λ q p2q 0 " " Č D p1q tF t s 0 ptu 0 s λ , wq ı Č ptu 0 s λ q p2q 0 `pθ, q ´∇ptu 0 s λ q p1q 0 " D p1q tF t s 0 ptu 0 s λ , wq ı `ptu 0 s λ q p1q 0 ˘¯`Ă V c pptu t s λ q p2q 0 q, where Ă V c pptu t s λ q p2q 0 q :" ż t 0 " Č D p1q tF τ,t s 0 ptu τ s 0 , wq ı ! `Φθ , Φ ˘`ptu τ s λ q p1q 0 , ptu τ s λ q p1q 0 , dB τ , dτ Φθ , Φ ˘`rD 1 tF τ s 0 sptu 0 s λ q p1q 0 , rD 1 tF τ s 0 sptu 0 s λ q p1q 0 , dB τ , dτ Φ0,2 θ , Φ 0,2 ˘pptu τ s λ q p1q 0 , dB τ , dτ q ´´ `Φ0,2 e i `ptu τ s λ q p1q 0 , ptu τ s λ q p1q 0 ˘˘e i , `ptu τ s 0 q ´1R `tu τ s 0 e i , tu τ s 0 θ `Φ0,2 e i `ptu τ s λ q p1q 0 , ptu τ s λ q p1q 0 ˘˘tu τ s 0 ¯dτ ) , Φ 0,2 e i `ptu τ s λ q p1q 0 , ptu τ s λ q p1q 0 ˘:"∇ p2q `ptu τ s λ q p1q 0 , ptu τ s λ q p1q 0 ˘H0 ptu τ s 0 , e i q
´∇p2q `rD p1q tF τ s 0 sptu 0 s λ q p1q 0 , rD p1q tF τ s 0 sptu 0 s λ q p1q 0 ˘H0 ptu τ s 0 , e i q `R`H 0 ptu τ s 0 , e i q, ptu τ s λ q p1q 0 ˘ptu τ s λ q p1q 0 ´R`H 0 ptu τ s 0 , e i q, rD p1q tF τ s 0 sptu 0 s λ q p1q 0 ˘rD p1q tF τ s 0 sptu 0 s λ q p1q 0 `2∇ `ptu τ s λ q p1q 0 ˘`H λ ˘p1q 0 ptu τ s 0 , e i q ``H λ ˘p2q 0 ptu τ s 0 , e i q.

Let rD p2q tF t s 0 p¨, wqs be the restriction of the second order tangent map of tF t s 0 on the space T ptus 0 ,ptus λ q p1q λ q T tus 0 Fp Ă M q. We can deduce from Corollary 4.20 and Corollary 4.23 that

´ptu t s λ q p1q 0 , ptu t s λ q p2q 0 ¯" rD p2q tF t s 0 p¨, wqs ´ptu 0 s λ q p1q 0 , ptu 0 s λ q p2q 0 Vc pptu t s λ q p1q 0 q, V c pptu t s λ q p2q 0 q ¯, `Č ptu t s λ q p1q 0 , Č ptu t s λ q p2q 0 ˘" r Č D p2q tF t s 0 p¨, wqs `Č ptu 0 s λ q p1q 0 , Č ptu 0 s λ q p2q 0 (4.43) `´Ă V c pptu t s λ q p1q 0 q, Ă V c pptu t s λ q p2q 0 q ¯.
Continuing the discussions in Lemma 4.18 and Lemma 4.21, we can derive the SDEs for the differentials ptu t s λ q pjq λ , 3 ď j ď k ´2, and their pull back Č ptu t s λ q pjq λ via the pθ, q-map, whose Itô forms involve t∇ pl 1 q pH λ q pj 1 q λ u j 1 ďj,l 1 `j1 ďj , t∇ pl 1 q R λ u l 1 ďj . We omit the details.

The SDEs of `rD plq tF t s λ p¨, wqs ˘pjq λ can be formulated as in Section 4.2 by analogy with the deterministic case. We only state the SDEs for the pl, jq " p1, 1q case using the reference connection of r g 0 , whose calculations can be done as in Lemma 4.5.

Lemma 4.24.

Let λ P p´1, 1q Þ Ñ g λ P M k pM q be a C k curve with k ě 4. Let λ Þ Ñ ptu 0 s λ , v λ 0 q P T Fp Ă M q be C 1 and write `tu t s λ , v λ t ˘:" ´tF t s λ ptu 0 s λ q, " D p1q tF t s λ ptu 0 s λ , wq ‰ v λ 0 ¯, pv λ t q p1q λ :" ∇ `ptu t s λ q p1q λ ˘vλ t .
i) The process pv λ t q p1q λ satisfies the Stratonovich SDE dpv λ t q p1q λ "∇ `pv λ t q p1q λ ˘Hλ ptu t s λ , ˝dB t q `∇p2q `vλ t , ptu t s λ q p1q λ ˘Hλ ptu t s λ , ˝dB t q `R`H λ ptu t s λ , eq, ptu t s λ q p1q λ ˘vλ t `∇pv λ t qpH λ q p1q λ ptu t s λ , ˝dB t q. ii) The process pθ, q tuts λ `pv λ t q p1q λ ˘satisfies the Stratonovich SDE d `pθ, qpv λ t q p1q λ " dpθ, q ´Hλ ptu t s λ , ˝dB t q, pv λ t q p1q λ ¯`∇ppv λ t q p1q λ q ´pθ, q `Hλ ptu t s λ , ˝dB t q ˘pθ, q ´∇p2q `vλ t , ptu t s λ q p1q λ ˘Hλ ptu t s λ , ˝dB t q `R`H λ ptu t s λ , ˝dB t q, ptu t s λ q p1q λ ˘vλ t pθ,

q ´∇pv λ t qpH λ q p1q λ ptu t s λ , ˝dB t q ¯.
iii) The Itô SDE of the process pθ, q tuts λ `pv λ t q p1q λ ˘is d `pθ, qpv λ t q p1q λ " dpθ, q ´Hλ ptu t s λ , dB t q, pv λ t q p1q λ ¯`∇ppv λ t q p1q λ q `pθ, q `Hλ ptu t s λ , dB t q ˘pθ, q ´∇p2q `vλ t , ptu t s λ q p1q λ ˘Hλ ptu t s λ , dB t q `R`H λ ptu t s λ , eq, ptu t s λ q p1q λ ˘vλ t pθ, q ´∇pv λ t qpH λ q p1q λ ptu t s λ , dB t q ¯`∇pH λ ptu t s λ , e i qq ! dpθ, q `Hλ ptu t s λ , e i q, pv λ t q p1q λ ∇p `vλ t ˘p1q λ q `pθ, q `Hλ ptu t s λ , e i q ˘˘`pθ, q `∇p2q `vλ t , ptu t s λ q p1q λ ˘Hλ ptu t s λ , e i q Ȓ`H λ ptu t s λ , e i q, ptu t s λ q p1q λ ˘vλ t ``θ, ˘`∇pv λ t q `Hλ ˘p1q λ ptu t s λ , e i q ˘) dt.

As before, the formulas in Lemma 4.24 can be simplified at λ " 0.

Corollary 4.25. Let λ P p´1, 1q Þ Ñ g λ P M k pM q be a C k curve with k ě 4 and let λ Þ Ñ ptu 0 s λ , v λ 0 q P T Fp Ă M q be C 1 . We retain all the notations in Lemma 4.24.

i) The process pθ, q tuts λ `pv λ t q p1q 0 ˘satisfies the Stratonovich SDE d `pθ, qppv λ t q p1q 0 q

˘" ´ ppv λ t q p1q 0 q ˝dB t , ptu t s 0 q ´1R `tu t s 0 ˝dB t , θppv λ t q p1q 0 q ˘tu t s 0 pθ,

q ´∇p2q `vλ t , ptu t s λ q p1q λ ˘Hλ ptu t s λ , ˝dB t q pθ, q ´R`H λ ptu t s λ , ˝dB t q, ptu t s λ q p1q λ ˘vλ t `∇pv λ t qpH λ q p1q λ ptu t s λ , ˝dB t q ¯.
ii) The Itô SDE of the process θ tuts λ `pv λ t q p1q 0 ˘in T FpR n q is d `θppv λ t q p1q 0 qq " ppv λ t q p1q 0 qdB t `Ricptu t s 0 θppv λ t q p1q 0 qqdt `Φθ pv λ t , ptu t s λ q p1q 0 , dB t , dtq `Φ1,1 θ pv λ t , ptu t s λ q p1q 0 , dB t , dtq,

where Φ θ p¨, ¨, dB t , dtq is given in (4.15) associated to tu t s 0 and3 Φ 1,1 θ pv 0 t , ptu t s λ q p1q 0 , dB t , dtq :" 2 `∇pv 0 t qpH λ q p1q 0 ptu t s 0 , e i q ˘ei dt `θ

´"H 0 ptu t s 0 , e i q, ∇pv 0 t qpH λ q p1q 0 ptu t s 0 , e i q ı¯d t.

The Itô SDE of the process tuts λ `pv λ t q p1q 0 ˘in T FpR n q is d ` ppv λ t q p1q 0 qq " ptu t s 0 q ´1R ´tu t s 0 dB t , tu t s 0 θppv λ t q p1q 0 q ¯tu t s 0 `ptu t s 0 q ´1R ´tu t s 0 e i , tu t s 0 ppv λ t q p1q 0 qe i ¯tu t s 0 dt `ptu t s 0 q ´1`∇ ptu t s 0 e i qR ˘´tu t s 0 e i , tu t s 0 θppv λ t q p1q 0 q ¯tu t s 0 dt `Φ pv λ t , ptu t s λ q p1q 0 , dB t , dtq `Φ1,1 pv λ t , ptu t s λ q p1q 0 , dB t , dtq,

where Φ p¨, ¨, dB t , dtq is given in (4.16) associated to tu t s 0 and Φ 1,1 pv 0 t , ptu t s λ q p1q 0 , dB t , dtq :" ´∇pv 0 t qpH λ q p1q 0 ptu t s 0 , dB t q ¯.

We can formulate pv λ t q p1q 0 and pθ, qpv λ t q p1q 0 by stochastic Duhamel principle. Corollary 4.26. We retain all the notations in Corollary 4.25.

i) The process pv λ t q p1q 0 has the expression pv λ t q p1q 0 " " D p1q tF t s 0 ptu 0 s 0 , wq ‰ ppv λ 0 q p1q 0 q `∇pptu 0 s λ q p1q 0 q " D p1q tF t s 0 ptu 0 s 0 , wq ‰ pv 0 0 q `Vc ppv λ t q p1q 0 q, where V c ppv λ t q p1q 0 q :" ż t 0 " D p1q tF τ,t s 0 ptu τ s 0 , wq ‰ ´∇pv 0 τ qpH λ q p1q 0 ptu τ s 0 , ˝dB τ q ¯.

ii) On T FpR n q, the process Č pv λ t q p1q 0 :" pθ, q tuts λ `pv λ t q p1q 0 ˘has the expression Č pv λ t q p1q 0 "r Č D p1q tF t s 0 ptu 0 s 0 , wqs Č pv λ 0 q p1q 0 `pθ, q ´∇pptu 0 s λ q p1q 0 q " D p1q tF t s 0 ptu 0 s 0 , wq

‰ pv 0 0 q Ă V c ppv λ t q p1q 0 q, where Ă V c ppv λ t q p1q 0 q " ż t 0 " Č D p1q tF τ,t s 0 ptu τ s 0 , wq ı ! `Φ1,1 θ , Φ 1,1 ˘pv 0 τ , ptu τ s λ q p1q 0 , dB τ , dτ q ´´ `∇pv 0 τ qpH λ q p1q 0 ptu τ s 0 , e i q ˘ei , 0 ¯dτ ) . ˘p1q λ .
Proof. By a comparison of the SDEs in Lemma 4.24 and Corollary 4.25 with those in Lemma 4.5, we can compute as in Corollary 4.6 to derive i) and ii). We note that for ii),

Ă V c ppv λ t q
p1q 0 q has an extra term ´´0, ptu τ s 0 q ´1R ´tu τ s 0 e i , tu τ s 0 θ `∇pv 0 τ qpH λ q p1q 0 ptu τ s 0 , e i q ˘¯tu τ s 0 ¯dτ, which turns out to be zero since θ `∇pv 0 τ qpH λ q p1q 0 ptu τ s 0 , e i q ˘is zero.

We are in a situation to state the norm estimations on the differential processes.

Proposition 4.27. Let λ Þ Ñ g λ P M k pM q be a C k curve with k ě 3. Let x P Ă M and λ Þ Ñ tu 0 s λ P O r g λ x p Ă
M q be a C k´2 curve in Fp Ă M q and let ttu t s λ u tPr0,T s be the solution to (4.41) with initial tu 0 s λ . i) For every q ě 1 and pl, jq with 1 ď l `j ď k ´2, there exist c pl,jq pqq depending on m, q and the norm bounds of t∇ pl 1 q pH λ q pj 1 q λ u j 1 ďj,l 1 `j1 ďl`j , t∇ pl 1 q R λ u l 1 ďl , and c pl,jq pqq depending on pl, jq, m, q and the norm bounds of t∇ pl 1 q R λ u l 1 ď1 , such that

E sup 0ďtătďT › › › › ´"D plq tF t,t s λ pu t , wq ‰ ¯pjq λ › › › › q ď c pl,jq pqqe c pl,jq pqqT , @T P R `. (4.44)
ii) Let T 0 ą 0. For each q ě 1, pl, jq with 1 ď l `j ď k ´2 and T ą T 0 , there exist c 1 pl,jq pqq, which depends on m, q and the norm bounds of t∇ pl 1 q pH λ q pj 1 q λ u j 1 ďj,l 1 `j1 ďl`j , t∇ pl 1 q R λ u l 1 ďl , and c 1 pl,jq pqq, which depends on pl, jq, m, q, T, T 0 and the norm bounds of t∇ pl 1 q R λ u l 1 ď1 , such that, for any x, y P Ă M ,

E P λ,x ,y,T sup 0ďtătďT › › › ›
´"D plq tF t,t s λ pu t , wq

‰ ¯pjq λ › › › › q ď c 1 pl,jq pqqe c 1 pl,jq pqqp1`d r g λ px,yqq . (4.45)
By using the SDEs formulated in Section 4.5, the estimation in (4.44) can be obtained using (4.24) and the estimation in (4.45) can be obtained using Lemma 4.15 (4.34) and (4.38). The proofs are similar to the second steps in the proofs of Proposition 4.11 and Proposition 4.16, respectively. We omit them.

The first differential of the heat kernels in metrics

Our main result in this section is a first step of the proof of Theorem 1.3.

Theorem 5.1. For any g 0 " g P M k pM q (k ě 3), there exist ι P p0, 1q and a neighborhood V g of g in M k pM q such that the following hold true for any

C k curve λ P p´1, 1q Þ Ñ g λ P V g . i) For any x P Ă M and T P R `, λ Þ Ñ p λ pT, x, ¨q is C 1 in C k,ι p Ă M q with pln p λ q p1q λ pT, x, yq `pln ρ λ q p1q λ pyq " φ 1 λ pT, x, yq, (5.1)
where ρ λ pyq " pdVol λ {dVol 0 qpyq and φ 1 λ is as in (5.16). ii) Let T 0 ą 0. For q ě 1 and l, 0 ď l ď k ´3, there are constants c λ,pl,1q pqq which depend on m, q, T , T 0 , }g λ } C l`3 and }X λ } C l`2 such that for all x P Ă M and T ą T 0 ,

(5.2)

› › ›∇ plq pln p λ q p1q λ pT, x, ¨q› › › L q ď c λ,pl,1q pqq. iii) The function x Þ Ñ ş Ă M pp λ q p1q
λ pT, x, yq r f pyq dVol r g λ pyq is continuous for any uniformly continuous and bounded r f P Cp Ă M q.

5.1. Strategy. We show Theorem 5.1 by describing the C 1 vector field z λ,1 T such that (1.7) holds true. Before that, let us recall some classical results for parabolic equations.

Let D Ă D 1 ˆD2 with D 1 being a bounded interval of R `and D 2 being a bounded connected open domain of Ă M . For g P M k pM q, consider the parabolic equation

(5.3) Lq :" p B Bt ´∆qq " r,
where ∆ is the r g-Laplacian on C 2 functions on Ă M and r is a continuous function on D. By a solution q to (5.3), we mean a function q on D which satisfies (5.3) and all the derivatives of which appear in Lq are continuous functions on D. Such a q can be smoother, depending on the regularities of both L and r. For instance, q is C 8 if both both L and r are C 8 . In our case, L varies C k´2 Hölder with respect to base points and q is mostly C k Hölder in general even in case r is smooth.

Lemma 5.2. ( [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]Theorem 11,p.74]) Let L be given in (5.3) which is C k´2 and Hölder continuous with exponent ι. Assume r in (5.3) is such that

D n
x D l t r, 0 ď n `2l ď k ´2, l ď l 1 , are Hölder continuous with exponent ι, where D b a means the b-th differential form with respect to the a-coordinate. If q is a solution to (5.2), then

D n
x D l t q, 0 ď n `2l ď k, l ď l 1 `1, exist and are Hölder continuous with exponent ι.

In particular, if r is ι-Hölder and q solves (5.3), Lemma 5.2 shows that all the differentials of q up to the second order (where B{Bt is considered as second order differential) exist and are ι-Hölder. The next lemma from [Fr] further shows these differentials have bounds completely determined by the bounds of q and r. For P " pτ, xq P D, define

d P " sup QPDpτ q dpP , Qq,
where Dpτ q is the intersection of the boundary of D with the half-space t ď τ . For a function f on D and any non-negative integers n, j and for ι P p0, 1q, define

|f | n,j " j ÿ l"0 N n,l rf s, |f | n,j`ι " |f | n,j `j ÿ 0 N n,l`ι rf s,
where N n,l rf s "

ÿ sup P PD ! d n`l P |D l x f pP q| ) ,
N n,l`ι rf s "

ÿ sup P ,QPD " mintd n`l`ι P , d n`l`ι Q u ¨|D l x f pP q ´Dl x f pQq| dpP , Qq ι * ,
and the summation is over all the differentials of order l.

Lemma 5.3. ([Fr, Theorem 1, p.92]) Let L be as in Lemma 5.2. There exists some geometric constant κ (which depends on ι, }g} C 1 ) such that if |r| 2,ι ă `8 and q is a bounded solution to (5.3) and all its derivatives appearing in Lq are ι Hölder, then |q| 0,2`ι ă κp|q| 0,0 `|r| 2,ι q.

(Both Lemma 5.2 and Lemma 5.3 were stated in [Fr] for domains in the Euclidean case. They apply to the manifold case since (5.3) can be treated locally in coordinate charts.)

A companion notion of a solution to a parabolic equation is a solution in the distribution sense. Recall that the distributions on the domain D are the linear continuous functionals on the test function space C 8 c pDq of compactly supported smooth functions on D. Given a distribution q on D, one can define its weak derivative of any order α, denoted by D α,w q, as a distribution on D by letting pD α,w qqpf q :" p´1q |α| qpD α f q, @f P C 8 c pDq. Any locally integrable function q P L 1 loc pDq can be identified with a distribution by letting qpf q :" ż D qf dt ˆdVol, @f P C 8 c pDq, and hence its weak derivatives of any order always exist. Let L be as in (5.3). The L distributional derivative of a distribution q on D will be denoted by L w q. Using mollifier and Lemma 5.3, we have the following classical result.

Lemma 5.4. Let g P C k pM q and let L be as in (5.3). Assume r P C 0,ι pDq for some ι ą 0 with |r| 2,ι ă 8. Then for any q P CpDq, L w q " r ùñ Lq " r.

As a corollary of the above lemmas, we have the following. f pyqφ 1 λ pT, x, yqp λ pT, x, yq dVol λ pyq.

Then, Theorem 5.1 i) holds true.

Proof. Let T P R `and x P Ă M . If (5.4) is true, then for any

f P C 8 c p Ă M q, ż Ă M
f pyq ´pλ pT, x, yqρ λ pyq ´p0 pT, x, yqρ 0 pyq ¯dVol 0 pyq " since the functions appearing on both sides are all continuous in y-variable and λ-variable.

ż λ 0 ż Ă M f pyqφ 1 r λ pT, x, yqp
Since λ Þ Ñ ρ λ is C k , (5.6) implies the existence of pp λ q p1q λ pT, x, yq for every y and

(5.7) pp λ q p1q λ pT, x, yq ¨ρλ pyq `pλ pT, x, yq ¨pρ λ q p1q λ pyq " φ 1 λ pT, x, yqp λ pT, x, yqρ λ pyq.

Then (5.7) implies that pp λ q p1q λ p¨, x, ¨q is a continuous function on R `ˆĂ M since we have the continuity in the pT, yq-coordinate of both p λ pT, x, yq and φ 1 λ pT, x, yq by assumption. Shrinking the neighborhood V g of g if necessary, we may assume there is ι ą 0 such that p λ pT, x, ¨q P C k,ι p Ă M q for all λ. Since it is a local problem, for pT, yq P R `ˆĂ M , we can also restrict ourselves to a bounded domain D containing pT, yq. Note that L λ p λ " 0. Lemma 5.3 implies |p λ pT, x, ¨q| 0,2`ι ă 8 on D. For each x P Ă M , since pp λ q p1q λ pT, x, yq is continuous in pT, yq, its weak derivatives in pT, yq of any order are well-defined. So (5.8) L λ,w pp λ q p1q λ pT, x, ¨q " pL λ q p1q,w p λ pT, x, ¨q " pL λ q p1q λ p λ pT, x, ¨q. We can handle the equation locally. Shrinking the domain D to D 1 if necessary, we deduce from |p λ pT, x, ¨q| 0,2`ι ă 8 on D that |pL λ q p1q λ p λ pT, x, ¨q| 2,ι ă 8 on D 1 . Since pp λ q p1q λ pT, x, ¨q is continuous, Lemma 5.4 implies that (5.8) holds true in the usual sense, i.e., (5.9) L λ pp λ q p1q λ pT, x, ¨q " ´pL λ q p1q λ p λ pT, x, ¨q.

Then we can apply Lemma 5.3 to conclude that |pp λ q p1q λ pT, x, ¨q| 0,2`ι ă 8 on D 1 and apply Lemma 5.2 to conclude that pp λ q p1q λ pT, x, ¨q P C k,ι pD 1 q. The norms of pp λ q p1q λ pT, x, ¨q in C k,ι pD 1 q are locally uniformly bounded in λ by using (5.9), Lemma 5.2 and Lemma 5.3. So the continuity of λ Þ Ñ pp λ q p1q λ pT, x, ¨q in Cp Ă M q is improved to the continuity in C k,ι p Ă M q.

For Theorem 5.1, it remains to find a candidate φ 1 λ pT, x, yq for Lemma 5.5. Let x P Ă M and let tu 0 s λ P O r g λ

x p Ă M q. Recall that the solution to the SDE

dtu t s λ " m ÿ i"1
H λ ptu t s λ , e i q ˝dB i t pwq

with initial value tu 0 s λ projects to be the Brownian motion tx t s λ on Ă M starting from x and the heat kernel function p λ pT, x, ¨q is just the density of the distribution of w Þ Ñ tx T s λ pwq under Q. Hence for any f P C 8 c p Ă M q, we have

ż Ă M
f pyqp λ pT, x, yq dVol λ pyq " E `f ptx T s λ pwqq ȃnd the equality continues to hold if we differentiate both sides in λ. Choose λ Þ Ñ tu 0 s λ to be a C k´2 curve. By Lemma 4.17, for almost all w and all t P R `, λ Þ Ñ tu t s λ pwq is C k´2 . By Proposition 4.27, the differentials ptu t s λ q pjq λ pwq, j ď k ´2, are L 1 integrable, uniformly in λ. Hence,

´E`f ptx T s r λ pwqq ˘¯p1q λ " E ´A∇ λ tx T s λ pwq pf ˝πqptu T s λ pwqq, ptu T s λ q p1q λ pwq E λ (5.10) " ż Ă M E ´@∇ λ y f pyq, Dπptu T s λ q p1q λ pwq D λ ˇˇtx T s λ pwq " y ¯¨p λ pT,
x, yq dVol λ pyq. (5.11) Note that (5.11) holds for every choice of tu 0 s λ P O r g λ p Ă M q at λ. For some technical considerations which we will mention later, we choose tu 0 s λ in O r g λ p Ă M q at random with a uniform distribution normalized to be probability 1 and then choose tu 0 s r λ " tu 0 s r λ ptu 0 s λ q ´1tu 0 s λ , r λ P p´1, 1q, where tu 0 s r λ is some fixed C k curve in Fp Ă M q with tu 0 s

r λ P O r g r λ p Ă M q.
Write E for the new expectation when the random choices of tu 0 s λ are taken into account. Then

´Ef ptx T s r λ q ¯p1q λ " ż Ă M E ´A∇ λ y f pyq, Dπptu T s λ q p1q λ pwq E λ
ˇˇtx T s λ pwq " y ¯¨p λ pT, x, yq dVol λ pyq.

(5.12)

For any

C k bounded vector field Y on Ă M , let (5.13) Φ 1 λ pY qpyq :" E ´Φ1 λ pY, wq ˇˇtx T s λ pwq " y ¯,
where (5.14) Φ 1 λ pY, wq :"

@ Y ptx T s λ pwqq, Dπptu T s λ q p1q λ pwq D λ .
We will show the linear functional Φ 1 λ is such that Φ 1 λ pY q is C 1 in y variable, from which we can deduce that (5.15) z λ,1 T pyq :" E ´Dπptu T s λ q p1q λ pwq ˇˇtx T s λ pwq " y īs a C 1 vector field on Ă M . Hence, we can apply the the classical integration by parts formula to (5.12) and compute that

´Epf ptx T s r λ pwqqq ¯p1q λ " ´ż Ă M f pyq ´pDiv λ z λ,1 T pyqq `@z λ,1 T pyq, ∇ λ ln p λ pT, x, yq D λ ¯pλ pT, x, yq dVol λ pyq.
This gives a candidate of φ 1 λ for Lemma 5.5 as (5.16) φ 1 λ pT, x, yq :" ´´pDiv λ z λ,1 T pyqq `@z λ,1 T pyq, ∇ λ ln p λ pT, x, yq

D λ ¯.
To justify that (5.16) is well-defined, we need to show the C 1 dependence of Φ 1 λ pY qpyq in y-variable. Let V be a smooth bounded vector field on Ă M and let tF s u sPR be the flow it generates. To compare Φ 1 λ pY qpF s pyqq with Φ 1 λ pY qpyq, our strategy is to extend every map F s on tx T s λ pwq, the endpoint of Brownian motion paths at time T , to be a map F s on Brownian paths up to time T . Let P λ x denote the product of the probability P λ

x with the uniform probability on O r g λ p Ă M q for the choice of tu 0 s λ . We will ensure the maps F s are such that P In order to show this differentiability in s, we will show that our one-parameter family of maps F s satisfy the following properties (see Proposition 5.23, Proposition 5.29 and Proposition 5.30), where all the integrals are taken with respect to P λ x conditioned on tx T s λ pwq " y.

i) P λ x
˝Fs is absolutely continuous with respect to P λ x and the Radon-Nikodyn derivative dP λ x ˝Fs {dP λ x is L q integrable for q ě 1, locally uniformly in the s parameter, ii) the differential of dP

λ x ˝Fs {dP λ x in s is E s T ¨pdP λ x
˝Fs {dP λ x q, where E s T is L q integrable for q ě 1, locally uniformly in the s parameter, and iii) ptu T s λ q p1q λ ˝Fs is differentiable in s with the differential stochastic process L q integrable for q ě 1, locally uniformly in the s parameter.

With these three properties, we will obtain and this differential is absolutely integrable, locally uniformly in the s parameter. Hence (5.21) is differentiable in the s parameter and we are allowed to take the differential inside the expectation sign. The uniform continuity of z λ,1 T pyq and Div λ z λ,1 T pyq in T and y will follow from (see the proof of Theorem 5.1 with k " 3) iv) the uniform continuity in T and y of

E `Φ1 λ ˇˇtx T s λ pwq " y ˘and E ˜`Φ 1 λ ˝Fs ¨dP λ x ˝Fs dP λ x ˘1 0 ˇˇˇˇt x T s λ pwq " y ¸.
The major part of the remaining subsections is devoted to the construction of F s and the verification of its properties i)-iv) mentioned above, which will conclude i) of Theorem 5.1. We will discuss Theorem 5.1 ii) and iii) in the last subsection.

Fix T ą 0. For each y P Ă M , we will construct a one parameter family of maps F s y on Brownian motion paths starting from y up to time T with F s y,x being its conditional map on paths that will arrive at x in time T . We will achieve this in two steps: one for the SDE description of F s y and the other for its existence by Picard's iteration argument. The desired map F s will be the collection of all F s

x T ,x . But, we need to justify the meaning of Φ 1 λ ˝Fs and dP λ x ˝Fs {dP λ x since Φ 1 λ and P λ

x are associated with the diffusion paths from x. This and the verification of i)-iv) will be done in Sections 5.4 and 5.5. Finally, in Section 5.6, we will show the assumption of Lemma 5.5 is satisfied and will give the estimations in (5.2) by an analysis of z λ,1

T pyq and Div λ z λ,1 T pyq using the SDE theory.

5.2.

A description of F s y . In this part, we fix T P R `. Let y P Ă M and β 0 P O r g y p Ă M q. For a smooth segment t Þ Ñ α t " pα t,1 , ¨¨¨, α t,m q P R m , t P r0, T s, with α 0 " o, let β " pβ t q tPr0,T s in O r g p Ă M q be the unique smooth segment with initial β 0 satisfying the differential equation

∇ D Bt β t " m ÿ i"1
Hpβ t , e i q ¨dα t,i dt .

In the language of Section 4, this means that β is the transportation (or development) of α in O r g p Ă M q with starting point β 0 using the parallelism differential form pθ, q. The Itô map

I β 0 : C 8 o pr0, T s, R m q Ñ C 8 y pr0, T s, Ă M q is given by (5.22) I β 0 pαq :" πpβq " β,
where π is the projection map from O r g y p Ă M q to Ă M . It is invertible since α for (5.22) can be uniquely determined by the equation

dα t dt " pβ t q ´1∇ D Bt β t ,
where β Ă O r g p Ă M q is the horizontal lift of β with initial value β 0 , i.e.,

∇ D Bt β t " Hpβ t , β ´1 t ∇ D Bt β t q.
For β P C 8 y pr0, T s, Ă M q, its I-preimage I ´1

β 0 pβq is called the anti-development of β in R m .
For a smooth segment (or curve) β " pβ t q tPr0,T s on Ă M , the classical parallel transportation map β t 1 ,t 2 of tangent vectors along the segments pβ t q tPrt 1 ,t 2 s (0 ď t 1 ď t 2 ď T ) is given by

β t 1 ,t 2 pvq " β t 2 ˝β´1 t 1 pvq, @v P T βt 1 Ă M ,
where β is a horizontal lift of β. This definition is independent of the horizontal lift chosen since if β 1 is another horizontal lift of β, then β 1 t " β t β 0 ´1β 1 0 for t P r0, T s and hence

β 1 t 2 ˝pβ 1 t 1 q ´1 " β t 2 β 0 ´1β 1 0 ˝`pβ 1 0 q ´1β 0 β ´1 t 1 ˘" β t 2 ˝β´1 t 1 .
The Itô map and the parallel transportation map can also be defined in the stochastic case. Call an O r g p Ă M q-valued continuous stochastic process β " pβ t q tPr0,T s horizontal if there exists a R m -valued continuous stochastic process α " pα t,1 , ¨¨¨, α t,m q tPr0,T s with α 0 " o such that β solves the Stratonovich SDE (5.23)

dβ t " m ÿ i"1
Hpβ t , e i q ˝dα t,i .

For a continuous stochastic process pβ t q tPr0,T s on Ă M , its horizontal lifts are those horizontal processes β in O r g p Ă M q projecting to it and its anti-developments in R m are those α satisfying (5.23) (cf. [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]). For a fixed y P Ă M and β 0 P O r g y p Ă M q, (5.23) is uniquely solvable for every semi-martingale α and the Itô map

I β 0 pαq :" πpβq " β
is well-defined. In the sprit of Section 4, I β 0 pαq is the projection process of a transportation (or development) of α in O r g p Ă M q using the parallelism differential form pθ, q. The oneto-one correspondence between α, β, and β for semi-martingales is discussed in [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF].

For a semi-martingale β " pβ t q tPr0,T s on Ă M , its horizontal lifts β are uniquely determined by the distribution of β 0 (cf. [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]Theorem 2.3.5]). Hence, for almost all w P Θ `, we can define a stochastic 'parallel transportation map' β t 1 ,t 2 of tangent vectors along the path segments pβ t pwqq tPrt 1 ,t 2 s (0 ď t 1 ď t 2 ď T ) by letting

β t 1 ,t 2 pvq :" β t 2 ˝β´1 t 1 pvq, @v P T βt 1 pwq Ă M .
As in the deterministic case, this definition is independent of the horizontal lift β chosen. Each β t 1 ,t 2 is an isometry between T βt 1 pwq Ă M and T βt 2 pwq Ă M with the inverse map p β t 1 ,t 2 q ´1pv 1 q :" β t 1 ˝β´1 t 2 pv 1 q, @v 1 P T βt 2 pwq Ă M .

Moreover, the parallel transportation maps β t 1 ,t 2 also satisfy the cocycle property

β t 1 ,t 3 " β t 3 ,t 2 ˝ β t 1 ,t 2 , @0 ď t 1 ď t 2 ď t 3 ď T.
Let V be a smooth bounded vector field on Ă M . For each y P Ă M , we obtain a smooth curve s Þ Ñ F s pyq, s P R, with dF s pyq ds " VpF s pyqq.

Let T ą 0 be fixed and let tpy t , t qu tPr0,T s be a stochastic pair which defines the r g-Brownian motion starting from y. The mapping w Þ Ñ py t pwqq tPr0,T s gives the distribution of Brownian paths up to time T in C y pr0, T s, Ă M q, where we use w to differ it from w for tx t s λ . We want to construct a one parameter family of mappings F s y on Brownian distributions py t q tPr0,T s so that y s t pwq :" `Fs y py r0,T s pwqq ˘ptq, @t P r0, T s, is differentiable in the s 'direction' for almost all w with initial restriction dy s 0 {ds " VpF s pyqq.

Choose a C 1 function s : r0, T s Ñ R `with sp0q " 1, spT q " 0 and lim For almost all w, we obtain a vector field along the paths y r0,T s pwq with Υ V,y ptq :" sptq ¨ 0,t pVpyqq, t P r0, T s, where

t 1 ,t 2 pvq :" t 2 ˝ ´1 t 1 pvq, @v P T yt 1 pwq Ă M , 0 ď t 1 ď t 2 ď T.
Our desired maps F s y on py t pwqq tPr0,T s are such that py s t pwqq tPr0,T s satisfy the equation (5.25) dy s t pwq ds " Υ V,y s ptq, where Υ V,y s ptq :" sptq ¨ s 0,t pVpy s 0 qq, t P r0, T s, and s denotes the parallel transportation map for the process y s . The length of tangent vectors remain unchanged under parallel transportations. Hence }Υ V,y s ptq} " sptq ¨}Vpy s 0 q} ď sptq supt}V pyq}u, which tends to zero of order pT ´tq as t Ñ T by our choice of s. So, if the processes y s exist, the ending points y s T remain in y T . Remark 5.6. In [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF], Hsu introduced a class of maps for the Brownian motion starting from some point y on a compact manifold: in our notation, Υ V,y s ptq " t p 9 hptqq, where h is a fixed R m valued curve from the Euclidean Cameron-Martin space, i.e., the completion of the space of smooth paths h : r0, T s Þ Ñ R m starting from the origin o with the Hilbert norm |h| " p ş 1 0 | 9 hptq| 2 q 1 2 . In his construction, the initial point y s 0 remain unchanged since h starts from o and hence the equations of all y s can be transferred back to R m using a single Itô map at y. In contrast, in our construction, our manifold is non-compact and we use a vector field V on the manifold instead of a Euclidean Cameron-Martin space element h to generate the random vector field Υ V,y s . Our ends y s T remain unchanged for almost all paths since sptq tends to zero as t goes to T ; while the initials y s 0 changes as s varies so the Itô transfer map of y s to R m also changes with s. The C 1 requirement of sptq is stronger than the L 2 integrability of the differentials of hptq. This is to guarantee that we can obtain a continuous version of the resulting process y s t (and all other related processes) in the parameter pt, sq (see Theorem 5.17), which is not true for general h.

We will solve the SDE (5.25) by identifying the anti-developments α s t " I ´1 s 0 py s t q using Picard's iteration method, where s 0 is the parallel transportation of 0 along the curve pF s pyqq sPR . In many places, the transferred equations using V only differ in notations from that for the case of h in [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]. But, technically, we have to write every steps in details since the construction is different, the footpoints of the Itô maps are shifting, and we need more regularity of y s t and also more information of the associated random structures. We first consider (5.25) for smooth paths. Let y P Ă M and β 0 P O r g y p Ă M q be fixed. For β " pβ t q P C 8 y pr0, T s, Ă M q, the equation (5.26)

Bβ s t Bs " Υ V,β s ptq :" sptq ¨ s 0,t pVpF s yqq, β 0 " β,
where s 0,t :" β s 0,t , is always solvable. Consider α s t " I ´1 β s 0 pβ s t q, where β s 0 is the parallel transportation of β 0 along the curve s Þ Ñ F s pyq. Then pBα s t {Bsq| s"0 differs from β ´1 t pΥ V,β 0 ptqq by an integral of curvature term, which can be determined by a standard calculation exactly as in [START_REF] Hsu | Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold[END_REF]Theorem 2.1]. We give the proof for completeness.

Lemma 5.7. Let V be a smooth bounded vector field on Ă M . For β P C 8 y pr0, T s, Ă M q, let β s be the solution to (5.26) and let β s be its horizontal lift in O r g p Ă M q with initial point β s 0 . i) The differential pα s t q 1 s :" Bα s t {Bs is given by

pα s t q 1 s " Υ V,α s ptq :" ż t 0 s 1 pτ qpβ s 0 q ´1`V pβ s 0 q ˘dτ ´ż t 0 K V,α s pτ q dα s τ ,
where

K V,α s pτ q " ż τ 0 pβ s ς q ´1R `βs ς p Bα s ς Bς q, Υ V,β s pςq ˘βs ς dς.
ii) The differential pβ

s t q 1 s :" ∇ D Bs β s t satisfies the equation $ & % ∇ D Bt `θ`p β s t q 1 s ˘˘" s 1 ptqpβ s 0 q ´1V pF s yq, ∇ D Bt ` `pβ s t q 1 s ˘˘" pβ s t q ´1R ´βs t p Bα s t
Bt q, β s t pβ s 0 q ´1sptqVpF s yq ¯βs t .

(5.27)

Proof. For iq, we have

B Bt ˆBα s t Bs ˙" B Bs ˆBα s t Bt ˙" ∇ D Bs ´θ`∇ D Bt β s t ˘¯.
Using the exterior differentiation formula in covariant derivative (cf. [GHL]) and the structure equation ( 4.3) for θ, we obtain

∇ D Bs ´θ`∇ D Bt β s t ˘¯" ∇ D Bt ´θ`∇ D Bs β s t ˘¯`dθ ´∇ D Bs β s t , ∇ D Bt β s t " s 1 ptqpβ s 0 q ´1pVpβ s 0 qq ´ `∇ D Bs β s t ˘p Bα s t Bt q.
We continue to compute that

`∇ D Bs β s t ˘" ż t 0 ∇ D Bτ ´ `∇ D Bs β s τ ˘¯dτ,
where, by using the exterior differentiation formula, Kerp q " HT Fp Ă M q and (4.4),

∇ D Bτ ´ `∇ D Bs β s τ ˘¯" ∇ D Bs ´ `∇ D Bτ β s τ ˘¯`d ´∇ D Bs β s τ , ∇ D Bτ β s τ " Ω ˆHpβ s τ , p Bα s τ Bτ qq, H ´βs τ , pβ s 0 q ´1rspτ qVpβ s 0 qs ¯" pβ s τ q ´1R ˆβs τ p
Bα s τ Bτ q, Υ V,β s pτ q ˙βs τ .

(5.28) For (5.27), the first equation is true by the construction. The second equation holds by (5.28) since Υ V,β s pτ q " β s τ pβ s 0 q ´1spτ qVpF s yq.

For every smooth segment α " pα t q tPr0,T s in R m , consider the associated flow maps tF α t 1 ,t 2 u 0ďt 1 ăt 2 ďT for the transportation of α to Ă M using the parallelism differential form pθ, q, where F α t 1 ,t 2 :

Fp Ă M q Ñ Fp Ă M q; β α t 1 Þ Ñ β α t 2 with pβ α t q tPrt 1 ,t 2 s solving the equation ∇ D Bt β α t " Hpβ α t , dα t dt q. Each F α t 1 ,t 2 is a C k´1 diffeomorphism since H is C k´1 and α is smooth. Let DF α t 1 ,t 2 be the tangent map of F α t 1 ,t 2 .
It can be read in the pθ, q-coordinate as follows. Lemma 5.8. ([Mal2, Proposition 3.2]) Let α " pα t q tPr0,T s Ă R m be a smooth segment.

For any t 1 P r0, T s and In the pθ, q-coordinate, we have

v α t 1 P T β α t 1 Fp Ă M q, let v α t :" " DF α t 1 ,t `βα τ , w ˘ı v α t
# d dt `θ`v α t ˘˘" `vα t ˘dαt dt , d dt ` `vα t ˘˘" pβ α t q ´1R ´βα t dαt dt , β α t `θ`v α t ˘˘¯β α t .
Let α s " pα s t q tPr0,T s be a one parameter family of smooth segments of curves in R m . For any t 1 , t 2 with 0 ď

t 1 ă t 2 ď T , s Þ Ñ DF s t 1 ,t 2 :" DF α s t 1 ,t 2 is said to be C 1 in s if the image curve s Þ Ñ rDF s t 1 ,t 2 sv s t 1 is C 1 for any C 1 curve s Þ Ñ v s t 1 P T β s t 1 Fp Ă M q.
Lemma 5.9. Let α s t " I ´1 β s 0 pβ s t q, where β s are given in Lemma 5.7. The tangent maps

pDF s t 1 ,t 2 q 0ďt 1 ăt 2 ďT are C 1 in s. Let s Þ Ñ v s t 1 P T β s t 1
Fp Ă M q be C 1 . Then the differential In the pθ, q-coordinate, we have

pv s t q 1 s :" ∇ D{Bs v s t ,
$ & % ∇ D Bt `θ`p v s t q 1 s ˘˘" `pv s t q 1 s ˘Bα s t Bt `θ ´f`v s t , pβ s t q 1 s ˘¯, ∇ D Bt ` `pv s t q 1 s ˘˘" pβ s t q ´1R ´βs t Bα s t Bt , β s t θ `pv s t q 1 s ˘¯β s t ` ´f`v s t , pβ s t q 1 s ˘¯.
Proof. Using (5.29), we obtain

∇ D Bt ∇ D Bs v s t " ∇ D Bs ∇ D Bt v s t `R ˆHpβ s t , Bα s t Bt q, pβ s t q 1 s ˙vs t " `∇ppv s t q 1 s qHpβ s t , ¨q˘B α s t Bt `f`v s t , pβ s t q 1 s ˘.
Using (5.30) and the structure equation ( 4.3) for θ, we continue to compute that

∇ D Bt `θ`p v s t q 1 s ˘˘" `∇ D Bt θ ˘`pv s t q 1 s ˘`θ ˆ`∇ppv s t q 1 s qHpβ s t , ¨q˘B α s t Bt ˙`θ ´f`v s t , pβ s t q 1 s ˘" dθ ˆHpβ s t , Bα s t Bt q, pv s t q 1 s ˙`θ ´f`v s t , pβ s t q 1 s ˘" `pv s t q 1 s ˘Bα s t Bt `θ ´f`v s t , pβ s t q 1 s ˘¯.
Similarly, using (5.30) and the structure equation (4.4) for , we obtain

∇ D Bt ` `pv s t q 1 s ˘˘" d ˆHpβ s t , Bα s t Bt q, pv s t q 1 s ˙` ´f`v s t , pβ s t q 1 s ˘" pβ s t q ´1R ˆβs t Bα s t Bt , β s t θ `pv s t q 1 s ˘˙β s t ` ´f`v s t , pβ s t q 1 s ˘¯.
We will solve (5.25) by identifying the anti-development of α s of y s in the set

A :" " α t " ż t 0 O τ dB τ `ż t 0 g τ dτ, t P r0, T s * ,
where O τ is an OpR m q valued F τ -adapted process, g τ is a R m valued F τ -adapted process with |g| ď Const. sup |V| and tF t u tPR `is the filtration of the Brownian motion in R m . We see that A is a complete infinite dimensional Banach space under the norm }α} 8,T :"

b }g} 2 8,T `}O} 2 8,T , where }g} 2 8,T " E sup tPr0,T s |g t | 2 , }O} 2 8,T " E sup tPr0,T s |O t | 2 .
Let V and s be as above. For α P A, let β be a horizontal process in O r g p Ă M q with projection β " I β 0 pαq on Ă M . For t P r0, T s, put Υ V,β ptq :" sptq ¨ β 0,t pVpβ 0 qq " β t β ´1 0 rsptqVpβ 0 qs. We define Υ V,α ptq :"

ż t 0 s 1 pτ qβ ´1 0 pVpβ 0 qq dτ ´ż t 0 K V,α pτ q ˝dα τ ,
where ˝denotes the Stratonovich stochastic integral, and (5.32) where Ric was defined in (4.12) and (5.33)

K V,α pτ q :" ż τ 0 β ´1 ς R `βς p˝dα ς q, Υ V,β pςq ˘βς . (5.31) Lemma 5.10. For α P A, the Itô forms of Υ V,α , K V,α are Υ V,α ptq " ż t 0 ! β ´1 0 rs 1 pτ qVpβ 0 qs ´Ric pΥ V,β pτ qq ) dτ ´ż t 0 xK V,α pτ q, dα τ y ": R V,α ptq ´ż t 0 xK V,α pτ q, dα τ y,
K V,α ptq " ż t 0 β τ ´1R `βτ dα τ , Υ V,β pτ q ˘βτ `ż t 0 β τ ´1 `∇pβ τ e i qR ˘`β τ e i , Υ V,β pτ q ˘βτ dτ.
Proof. Using Itô's formula, we can identify the Itô integral expression of Υ V,α as Υ V,α ptq "

ż t 0 s 1 pτ qβ ´1 0 pVpβ 0 qq dτ ´ż t 0 xK V,α pτ q, dα τ y ´1 2 ż t 0 xdK V,α pτ q, ˝dα τ y.
Let α t " α t,1 e 1 `¨¨¨`α t,m e m , where te 1 , ¨¨¨, e m u is the standard orthogonal base of R m . Since α P A, we see that x˝dα t,i , ˝dα t,i y " 2dt. So, using (5.31), we obtain 1 2

ż t 0 xdK V,α pτ q, ˝dα τ y " 1 2 ż t 0 xβ ´1 τ R `βτ p˝dα τ q, Υ V,β pτ q ˘βτ , ˝dα τ y " ż t 0 β ´1 τ R `βτ e i , Υ V,β pτ q ˘βτ e i dτ " ż t 0 Ric pΥ V,β pτ qq dτ.
The Itô integral expression for K V,α ptq can be obtained similarly using Itô's formula.

We want to solve (5.25) with y 0 being the Brownian motion on Ă M starting from y.

Lemma 5.11. Let V be a smooth bounded vector field on Ă M with the associated flow tF s u sPR . For y P Ă M , let p s 0 P O r g F s y p Ă M qq sPR be a solution to d s 0 {ds " Hp s 0 , p s 0 q ´1VpF s yqq.

i) Let α s " ş 0 O s τ dB τ `ş¨0 g s τ dτ P A be a one parameter family of stochastic processes with α 0 t " B t . Then α s solves dα s t pwq{ds " Υ V,α s ptq iff

O s τ " Id ´ż s 0 K V,α  pτ qO  τ d, (5.34) g s τ " O s τ ż s 0 rO  τ s ´1 p  0 q ´1 " s 1 pτ qVpF  yq ‰ ´Ric `  τ p  0 q ´1rspτ qVpF  yqs ˘( d. (5.35)
ii) Let α s be as in iq and let s be its horizontal lift in O r g p Ă M q with initial s 0 . Then s is differentiable in s iff the following SDE is uniquely solvable with initial p s 0 q 1 s : " dθpY s t q " `Y s t ˘˝dα s t `˝dΥ V,α s , d pY s t q " p s t q ´1R ` s t ˝dα s t , s t θpY s t q ˘ s t .

(5.36)

iii) Let α s , s be as in i), ii). Then s Þ Ñ y s " I s 0 pα s q has the differential process Υ V,y s .

Proof. By analogy with the deterministic case (Lemma 5.7), we have y s solves (5.25) iff α s solves dα s t pwq{ds " Υ V,α s ptq, which means

α s t ´α0 t " ż s 0 ż t 0 p  0 q ´1rs 1 pτ qVpF  yqs dτ d ´ż s 0 ż t 0 xK V,α  pτ q, dα  τ y d ´ż s 0 ż t 0 Ric `  τ p  0 q ´1rspτ qVpF  yqs ˘dτ d " ż s 0 ż t 0 pu  T q ´1rs 1 pτ qVpF  yqs dτ d ´ż s 0 ż t 0 Ric `  τ p  0 q ´1rspτ qVpF  yqs ˘dτ d ´ż s 0 ż t 0 K V,α  pτ qg  τ dτ d ´ż s 0 ż t 0 K V,α  pτ qO  τ dB τ d.
Note that α 0 t " B t and hence O 0 " Id, g 0 " 0. So a comparison of the above expression with the the assumption that α s t " ş t 0 O s τ dB τ `şt 0 g s τ dτ gives (5.34) and

g s τ " ż s 0 p  0 q ´1rs 1 pτ qVpF  yqs d ´ż s 0 Ric `  τ p  0 q ´1rspτ qVpF  yqs ˘d ´ż s 0 K V,α  pτ qg  τ d.
Hence by the variation of constants method (i.e., Duhamel's principle), we obtain (5.35).

Let α s be as in i) which solves dα s t pwq{ds " Υ V,α s ptq. Then s is differentiable in s iff the following SDE is solvable with initial p s 0 q 1 s : dY s t " `∇pY s t qH ˘p s t , ˝dα s t q `Hp s t , p˝dα s t q 1 s q. (5.37) Writing (5.37) in the pθ, q-coordinate, we have dpθpY s t qq " dθp˝d s t , Y s t q `θ`H p s t , p˝dα s t q 1 s q ˘" ωpY s t q ˝dα s t `˝dΥ V,α  , dp pY s t qq " Ω pHp s t , ˝dα s t q, Hp s t , θpY s t qqq " p s t q ´1R ` s t ˝dα s t , s t θpY s t q ˘ s t .

Let α s , s be such that i), ii) hold true. For iii), it suffices to check the equality ∇ D{Bs pπp s qq " Υ V,y s . Let Z  t :" θ `p s t q 1  ˘. By (5.36),

Z  t " Υ V,α  ptq ´ΥV,α  p0q `ż t 0 ˆż τ 0 p  τ 1 q ´1R `  τ 1 ˝dα  τ 1 ,  τ 1 Z  τ 1 ˘  τ 1 ˙˝dα s τ .
Write Z  t :" Z  t ´şt 0 p  0 q ´1rs 1 pτ qVpF  yqs dτ . Then we have

Z  t " ż t 0 ˆż τ 0 p  τ 1 q ´1R `  τ 1 ˝dα  τ 1 ,  τ 1 Z  τ 1 ˘  τ 1 ˙˝dα s τ .
Using Itô's formula for | ¨|2 " x¨, ¨y or the isometry property of Brownian motion, we can find some constant Cps 0 , T q depending on R, s 0 , T such that EppZ  t q 2 q ď Cps 0 , T q

ż t 0 EppZ  τ q 2 q dτ.
This gives Z  t " 0 by Gronwall's Lemma (see Lemma 5.16). Thus py s q 1 s " Υ V,y s . Corollary 5.12. Let s be as in ii) of Lemma 5.11. Then Y s " p s q 1 s is given by " dθpY s t q " s 1 ptqp s 0 q ´1V pF s yq dt, d pY s t q " p s t q ´1R ` s t ˝dα s t , sptq s t p s 0 q ´1V pF s yq ˘ s t ,

(5.38)

whose Itô form is $ & % dθpY s t q " s 1 ptqp s 0 q ´1V pF s yq dt, d pY s t q " p s t q ´1R ` s t dα s t , sptq s t p s 0 q ´1V pF s yq ˘ s t `p s t q ´1p∇p s t e i qRq ` s t e i , sptq s t p s 0 q ´1V pF s yq ˘ s t dt.

(5.39)

Proof. Note that s is a horizontal lift of y s . Reporting this and py s q 1 s " Υ V,y s in (5.36) shows (5.38). Then (5.39) follows by applying the Itô formula.

For α " pα t,1 , ¨¨¨, α t,m q P A, consider the associated flow maps tF α t 1 ,t 2 u 0ďt 1 ăt 2 ďT , where

F α t 1 ,t 2 : Fp Ă M q Ñ Fp Ă M q; β α t 1 Þ Ñ β α t 2 ,
, Q s t 1 q P T β s Fp Ă M q which is C 1 in s, rDF s t 1 ,t 2 spυ s t 1 , Q s t 1 q is also C 1 in s.
The following can be formulated using Lemma 5.13 and Itô's formula by analogy with Lemma 5.9.

Lemma 5.14. Let α s , y s and s be as in Lemma 5.11. Then tDF s t 1 ,t 2 u 0ďt 1 ăt 2 ďT are C 1 in s iff for any v s t 1 P T s t 1

Fp Ă M q C 1 in s, there is a unique pυ s t q tPrt 1 ,t 2 s , continuous in pt, sq with υ s t 1 " ∇ D{Bs v s t 1 , that solves the SDE

dυ s t " `∇pυ s t qH ˘p s t , ˝dα s t q `f`v s t , p s t q 1 s ˘, (5.41)
where f `vs t , p s t q 1 s ˘"∇ p2q `vs t , p s t q 1 s ˘Hp s t , ˝dα s t q `∇pv s t qH ` s t , ˝dΥ V,α s ptq Ȓ `Hp s t , ˝dα s t q, p s t q 1 s ˘vs t .

In the pθ, q-coordinate, (5.41) is where f I `vs t , p s t q 1 s ˘is f `vs t , p s t q 1 s ˘with ˝dα s t replaced by the Itô infinitesimal dα s t ,

" d `θ`
f θ A `vs t , p s t q 1 s ˘"2 `f`v s t , p s t q 1 s , e i ˘˘e i dt `θ `"Hp s t , e i q, f `vs t , p s t q 1 s , e i ˘‰˘d t, f A `vs t , p s t q 1 s ˘"2p s t q ´1R
` s t e i , s t θ `f pv s t , p s t q 1 s , e i q ˘˘ s t dt ` `"Hp s t , e i q, f `vs t , p s t q 1 s , e i ˘‰˘d t, f `vs t , p s t q 1 s , e i ˘"∇ p2q `vs t , p s t q 1 0 ˘Hp s t , e i q`R `Hp s t , e i q, p s t q 1 s ˘vs t `∇pv s t qH

` s t , K V,α s ptqe i ˘.
5.3. The existence of F s y . In this part, we prove the existence of the mapping y Þ Ñ F s y pyq. By Lemma 5.11, it suffices to solve dα s t pwq{ds " Υ V,α s ptq in A with α 0 " B. We will do this using the classical Picard method as in [START_REF] Hsu | Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold[END_REF]Theorem 3.1]. In the meanwhile, we will also show the existence of the differential processes of s and DF s t 1 ,t 2 in s. The tool we will use to obtain a continuous version of a two-parameter process is Kolmogorov's criterion.

Lemma 5.15. (cf. [START_REF] Kunita | Stochastic flows and stochastic differential equations[END_REF]Theorem 1.4.1]) Let tY s t pwqu tPr0,T s,sPr´s 0 ,s 0 s be a one parameter family of random processes on a complete manifold. Suppose there are positive constants 5, 5 1 , 5 2 , with 5 1 , 5 2 ą 2, and C 0 p5q such that for all t, t 1 P r0, T s and s, s 1 P r´s 0 , s 0 s,

E " ˇˇY s t ´Ys 1 t 1 ˇˇ5  ď C 0 p5q ´|t ´t1 | 5 1 `|s ´s1 | 5 2 ¯,
then Y s t has a continuous modification with respect to the parameter pt, sq.

Besides Burkholder's inequality (Lemma 4.7), another useful tool to estimate the L qnorm of stochastic integrals is Gronwall's lemma: Lemma 5.16. (cf. [START_REF] Elworthy | Stochastic Differential Equations on Manifolds[END_REF]p. 13]) Let φ, φ 1 be real valued Lebesgue integrable functions on the interval r0, ss such that for some C ą 0, φpq ď φ 1 pq `C ż  0 φp 1 q d 1 , @ P r0, ss.

Then

φpq ď φ 1 pq `C ż  0
e Cp´ 1 q φ 1 p 1 q d 1 , for almost all  P r0, ss.

We are in a situation to state the existence theorem of the maps F s y .

Theorem 5.17. Let V be a bounded smooth vector field on Ă M and let tF s u sPR be the flow it generates. For y P Ă M , let p s 0 P O r g F s y p Ă M qq sPR with d s 0 {ds " Hp s 0 , p s 0 q ´1VpF s yqq be a fixed horizontal lift of the smooth curve pF s yq sPR .

i) There exists a unique family of stochastic processes α s P A such that for almost all w, s Þ Ñ α s pwq is differentiable with

(5.44) α s t pwq " w `ż s 0 Υ V,α  pt, wq d, @t P r0, T s.

The process Υ V,α s ptq has a continuous modification in the parameter pt, sq. ii) Let s P O r g p Ă M q be a horizontal lift of y s with initial s 0 . There exists a one parameter family of F t -adapted stochastic processes pY s t q tPr0,T s with Y s t pwq P T s t pwq pO r g p Ă M qq for almost all w, which satisfies ∇ D Bs s t pwq " Y s t pwq, @t P r0, T s.

The process Y s t has a continuous modification in the parameter pt, sq.

iii) Let y s " I s 0 pα s q. Then s Þ Ñ y s pwq is differentiable for almost all w with

(5.45) ∇ D Bs y s t pwq " Υ V,y s pt, wq, @t P r0, T s.

The process Υ V,y s ptq has a continuous modification in the parameter pt, sq. iv) For almost all w, pDF s t 1 ,t q 0ďt 1 ătďT are C 1 in s. For v s t 1 P T s t 1

Fp Ă M q C 1 in s, the process v s t " rDF s t 1 ,t sv s t 1 is differentiable in s and the differential process υ s t has a continuous modification in the parameter pt, sq.

Proof. For simplicity, we will use C to denote a constant depending on }g} C 3 and the norm bound of V and use Cp¨q to indicate the extra coefficients it depends on, for instance, Cps 0 , T q means C also depends on s 0 , T . These constants C may vary from line to line.

We first show i). For any s 0 P R `, we use (5.34), (5.35) for Picard's iteration and show the iteration converges to a one-parameter family of processes α s (s ď s 0 ) in norm } ¨}8,T . Let g s,0 " 0, O s,0 " Id, α s,0 " B and let s,0 be the horizontal lift of I s 0 pBq in O r g p Ă M q with s,0 0 " s 0 . Assume g s,n´1 , O s,n´1 and α s,n´1 are obtained for some n P N. We write s,n´1 for the horizontal development of α s,n´1 in O r g p Ă M q with s,n´1 0 " s 0 and put

K V,α s,n´1 ptq " ż t 0 p s,n´1 τ q ´1R ` s,n´1 τ dα s,n´1 τ , s,n´1 τ p s 0 q ´1rspτ qVpF s yqs ˘ s,n´1 τ `ż t 0 p s,n´1 τ q ´1p∇p s,n´1 τ e i qRq ` s,n´1 τ e i , s,n´1 τ p s 0 q ´1rspτ qVpF s yqs ˘ s,n´1 τ dτ,
R V,α s,n´1 ptq " p s 0 q ´1rs 1 ptqVpF s yqs ´Ric

´ s,n´1 t p s 0 q ´1rsptqVpF s yqs ¯.

Then define g s,n , O s,n , α s,n as the processes determined by the following SDEs:

$ ' ' ' ' & ' ' ' ' % O s,n t " Id ´şs 0 K V,α ,n´1 ptqO ,n t d, g s,n t " O s,n t ş s 0 rO ,n t s ´1R V,α ,n´1 ptq d, α s,n t " ş t 0 O s,n τ dB τ `şt 0 g s,n τ dτ.
When n " 1, the definitions of g s,0 , O s,0 , g s,1 and O s,1 show that

O s,1 t ´Os,0 t " ´ż s 0 K V,α ,0 ptqO ,1 t d, g s,1 t ´gs,0 t " O s,1 t ż s 0 rO ,1 t s ´1R V,α ,0 ptq d.
Abbreviate } ¨}8,T as } ¨}. There is some C such that

}K V,α ,0 } 2 ď 2E sup tPr0,T s ˇˇˇż t 0 p s,0 τ q ´1R
` s,0 τ dB τ , s,0 τ p s 0 q ´1rspτ qVpF s yqs

˘ s,0 τ ˇˇˇ2 `CT 2 , ď 4E ˇˇˇż T 0 p s,0 τ q ´1R
` s,0 τ dB τ , s,0 τ p s 0 q ´1rspτ qVpF s yqs

˘ s,0 τ ˇˇˇ2 `CT 2 , ď 4E ż T 0 ˇˇp s,0 τ q ´1R
` s,0 τ e i , s,0 τ p s 0 q ´1rspτ qVpF s yqs

˘ s,0 τ ˇˇ2 dτ `CT 2 ď CpT `T 2 q,
where the second inequality holds by Doob's inequality of sub-martingales and the third inequality holds by Lemma 4.7. Hence there is some CpT q such that }O s,1 ´Os,0 } ď

ż s 0 }K V,α ,0 } ds ď CpT qs.
There also exists some C such that }g s,1 ´gs,0 } ď Cs since R V,α ,0 is bounded. So we obtain some C 0 pT q such that }α s,1 ´αs,0 } ď C 0 pT qs.

If we can further find some constant C 1 pT q such that }α s,n ´αs,n´1 } ď C 1 pT q ż s 0 }α ,n´1 ´α,n´2 } d, (5.46) we will obtain }α s,n ´αs,n´1 } ď 1 n! pC 0 pT q `C1 pT qq n s n , which will imply the existence of the limits

g s " lim nÑ`8 g s,n , O s " lim nÑ`8 O s,n .
Then α s t " ş t 0 O s τ dB τ `şt 0 g s τ dτ will be our desired process for i) by Lemma 5.11. For (5.46), let us analyze }O s,n ´Os,n´1 } and }g s,n ´gs,n´1 }. Since each O s,n is OpR m q valued and is invertible, we have

O s,n ´Os,n´1 " O s,n `Id ´pO s,n q ´1O s,n´1 " ´Os,n ż s 0 d ds " pO s,n q ´1O s,n´1 ‰ ˇˇˇs " d " ´Os,n ż s 0 ˜d ds " pO s,n q ´1‰ ˇˇˇs " O ,n´1 `pO ,n q ´1 d ds " O s,n´1 ‰ ˇˇˇs " ¸d.
By the inductive defining equations of O s,n´1 , O s,n , we obtain

d ds " O s,n´1 ‰ ˇˇˇs " " ´KV,α ,n´2 O ,n´1 , d ds " pO s,n q ´1‰ ˇˇˇs " " ´pO ,n q ´1 d ds rO s,n s ˇˇˇs " pO ,n q ´1 " pO ,n q ´1K V,α ,n´1 .
Hence O s,n ´Os,n´1 " ´Os,n ż s 0 pO ,n q ´1 `KV,α ,n´1 ´KV,α ,n´2 ˘O,n´1 d.

Using (5.33), we conclude that there are some constants C, C 1 such that Hence }g s,n ´gs,n´1 } ď }paq} `}pbq} `}pcq}.

}O s,n ´Os,n´1 } ď C ż s 0 › › K V,α ,n´1 ´KV,α ,n´2 › › d ď C 1 ż s 0 }α ,n´1
Since V is bounded on Ă M and s is C 1 on r0, T s, R V,α ,n´1 ptq is also bounded. So there are some constants C, C 1 such that

}paq} ď Cs 0 › › O s,n ´Os,n´1 › › ď C 1 s 0 ż s 0 }α ,n´1 ´α,n´2 } d, }pbq} ď C ż s 0 › › O ,n´1 ´O,n › › d ď C 1 ż s 0 }α ,n´1 ´α,n´2 } d.
For pcq, we also have

}pcq} ď C ż s 0 › › R V,α ,n´1 ´RV,α ,n´2 › › d ď C 1 ż s 0 › › ,n´1 ´ ,n´2 › › d,
where the last norm is measured using the distance function on O r g p Ă M q. Recall that

,n t " ,n 0 `ż t 0 m ÿ i"1
Hp ,n τ , e i q ˝dα ,n;i τ , 0 ď t ď T, (5.48) where α ,n;i , i " 1, ¨¨¨, m, denotes the i-th component of α ,n . Embedding O r g p Ă M q into some higher dimensional Euclidean space R l and extending all Hp¨, e i q to a small tube neighborhood of O r g p Ă M q, we can consider (5.48) as a Euclidean SDE and compare › › ,n´1 ´ ,n´2 › › with }α ,n´1 ´α,n´2 } using the Euclidean norm. By (5.48),

,n´1 t ´ ,n´2 t " ż t 0 `Hp ,n´1
τ , e i q ´Hp ,n´2 τ , e i q ˘˝pO ,n´1 τ

dB τ q i `ż t 0 Hp ,n´2 τ , e i q ˝ppO ,n´1 τ ´O,n´2 τ qdB τ q i `ż t 0 `Hp ,n´1
τ , e i q ´Hp

¸.

For pIq t , we can consider its Itô form and then apply Doob's inequality of sub-martingales and Lemma 4.7, which gives

E sup tPr0, r ts |pIq t | 2 ď CE ż r t 0 ˇˇ ,n´1 τ ´ ,n´2 τ ˇˇ2 dτ ď C ż r t 0 E sup tPr0,τ s ˇˇ ,n´1 t ´ ,n´2 t ˇˇ2 dτ.
The same argument shows there is some C such that

E sup tPr0, r ts |pIIq t | 2 ď CE ż r t 0 ˇˇO ,n´1 τ ´O,n´2 τ ˇˇ2 dτ ď CT › › α ,n´1 ´α,n´2 › › 2 .
Note that }Hp¨, e i q} and |g s,n´1 | (for all s and n) are bounded. Hence

E sup tPr0, r ts |pIIIq t | 2 ď C r tE ż r t 0 ˇˇ ,n´1 τ ´ ,n´2 τ ˇˇ2 dτ ď CT ż r t 0 E sup tPr0,τ s ˇˇ ,n´1 t
´ ,n´2 t ˇˇ2 dτ.

For pIVq t , we have

E sup tPr0, r ts |pIVq t | 2 ď C r t 2 › › g ,n´1 ´g,n´2 › › ď CT 2 › › α ,n´1 ´α,n´2 › › 2 .
Altogether, there are some constants C 2 pT q, C 3 pT q such that

E sup tPr0, r ts ˇˇ ,n´1 t ´ ,n´2 t ˇˇ2 ď C 2 pT q › › α ,n´1 ´α,n´2 › › 2 `C3 pT q ż r t 0 E sup tPr0,τ s ˇˇ ,n´1 t ´ ,n´2 t ˇˇ2 dτ.
Applying Lemma 5.16, we obtain some constant CpT q independent of  such that

› › ,n´1 ´ ,n´2 › › ď CpT q › › α ,n´1 ´α,n´2 › › . So, }pcq} ď C ż s 0 › › ,n´1 ´ ,n´2 › › d ď CpT q ż s 0 › › α ,n´1 ´α,n´2 › › d.
Putting together the estimations of }paq}, }pbq} and }pcq}, we conclude that

› › g s,n ´gs,n´1 › › ď CpT q ż s 0 › › α ,n´1 ´α,n´2 › › d.
This and (5.47) imply (5.46). Hence the limit lim nÑ8 α s,n " ż 0 O s τ dτ `ż 0 g s τ dτ ": α s exists and α s satisfies the equation (5.44) by i) of Lemma 5.11.

The tα s u obtained by the above iteration is the the unique parameter of processes in A satisfying (5.44). Assume tr α s u Ă A is another parameter of processes solving (5.44). Then, by using (5.34), (5.35) for α s and r α s , respectively, the above argument shows that (5.46) holds true by replacing α s,n , α s,n´1 by α s , r α s , respectively, for all s, i.e.,

}α s ´r α s } ď CpT q ż s 0 }α  ´r α  } d.
This implies α s " r α s by Gronwall's lemma.

We proceed to show E ˇˇΥ V,α s 1 pt 1 q ´ΥV,α s ptq ˇˇ5 ď Cp5, s 0 , T q ´|t 1 ´t| 1 2 5 `|s 1 ´s| 5 (5.49)

for any 5 ą 4, t, t 1 P r0, T s and s, s 1 P r´s 0 , s 0 s. This, by applying Lemma 5.15, will imply that Υ V,α s ptq has a continuous modification in the parameter pt, sq. Without loss of generality, we assume t ă t 1 . Using (5.32) and (4.22), we compute that

E ˇˇΥ V,α s 1 pt 1 q ´ΥV,α s ptq ˇˇ5 ď 5 5´1 ˜Eˇˇż t 1 t R V,α s 1 pτ q dτ ˇˇ5 `Eˇˇż t 1 t xK V,α s 1 pτ q, dα s 1 τ y ˇˇ5 `Eˇˇż t 0 pR V,α s 1 ´RV,α s qpτ q dτ ˇˇ5 ´Eˇˇż t 0 xpK V,α s 1 ´KV,α s qpτ q, dα s 1 τ y ˇˇ5 `Eˇˇż t 0 xK V,α s pτ q, dpα s 1 τ ´αs τ qy ˇˇ5 ":
5 5´1 `pdq `peq `pfq `pgq `phq ˘.

To conclude (5.49), we will show pdq, peq ď Cp5, s 0 , T q|t 1 ´t| 1 2 5 and pfq, pgq, phq ď Cp5, s 0 , T q|s 1 ´s| 5 . (5.50) Clearly, pdq ď C 5 ˇˇt 1 ´tˇˇ5 ď pCT q 5 |t 1 ´t| 1 2 5

for some C which bounds |R V,α  |. For peq, we have

2 1´5 peq ď E ˇˇż t 1 t xK V,α s 1 pτ q, O s 1 τ dB τ y ˇˇ5 `Eˇˇż t 1 t xK V,α s 1 pτ q, g s 1 τ y dτ ˇˇ5 ": peq 1 `peq 2 .
By Lemma 4.7, with the constant C 1 p5q there,

peq 1 ď C 1 p5qE ˇˇż t 1 t |K V,α s 1 pτ q| 2 dτ ˇˇ1 2 5 ď C 1 p5q `ż t 1 t Ep|K V,α s 1 pτ q| 5 q dτ ˘¨|t 1 ´t| 1 2 5´1 .
Using (4.22) and Lemma 4.7, it is easy to deduce that

Ep|K V,α s 1 pτ q| 5 q ď 3 5´1 ˜E ˇˇˇż τ 0 p s 1 τ 1 q ´1R ´ s 1 τ 1 O s 1 τ 1 dB τ 1 , s 1 τ 1 p s 1 0 q ´1rspτ 1 qVpF s 1 yqs ¯ s 1 τ 1 ˇˇˇ5 `E ˇˇˇż τ 0 p s 1 τ 1 q ´1R ´ s 1 τ 1 g s 1 τ 1 dτ 1 , s 1 τ 1 p s 1 0 q ´1rspτ 1 qVpF s 1 yqs ¯ s 1 τ 1 ˇˇˇ5 `E ˇˇˇż τ 0 p s 1 τ 1 q ´1p∇p s 1 τ 1 e i qRq ` s 1 τ 1 e i , s 1 τ 1p s 1 0 q ´1rspτ 1 qVpF s 1 yqs ˘ s 1 τ 1 dτ 1 ˇˇˇ5 ḑ 3 5´1 Cpτ 1 2 5 `τ 5 q. (5.51)
Hence there is a constant Cp5, T q such that peq 1 ď Cp5, T q|t 1 ´t| 1 2 5 . Since |g s τ | is bounded by some constant depending on s 0 and sup |V|, using Hölder's inequality and the estimation in (5.51), we obtain

peq 2 ď `ż t 1 t
Ep|K V,α s 1 pτ q| 5 q dτ ˘¨|t 1 ´t| 5´1 ď Cp5, T q|t 1 ´t| 5 .

Thus, peq ď 2 5´1 ppeq 1 `peq 2 q ď 2 5´1 pT 1 2 5 `1qCp5, T q|t 1 ´t| 1 2 5 " Cp5, T q|t 1 ´t| 1 2 5 . Using Hölder's inequality and Burkholder's inequality, the conclusion in (5.50) for pfq, pgq and phq holds if

E ˇˇpR V,α s 1 ´RV,α s qpτ q ˇˇ5 , E ˇˇpK V,α s 1 ´KV,α s qpτ q ˇˇ5 , E ˇˇα s 1 τ ´αs τ ˇˇ5 ď Cp5, s 0 , T q|s 1 ´s| 5 ,
which can be further reduced to verifying

E ˇˇO s 1 τ ´Os τ ˇˇ5 , E ˇˇg s 1 τ ´gs τ ˇˇ5 , E ˇˇ s 1 τ ´ s τ ˇˇ5 ď Cp5, s 0 , T q|s 1 ´s| 5 .
By (5.34) and (5.51), there is some constant Cp5, T q such that

E ˇˇO s 1 τ ´Os τ ˇˇ5 " E ˇˇˇˇż s 1 s K V,α  pτ qO  τ d ˇˇˇˇ5 ď ˇˇˇˇż s 1 s Ep|K V,α s 1 pτ q| 5 q dτ ˇˇˇˇ¨| s 1 ´s| 5´1
ď Cp5, T q|s 1 ´s| 5 . (5.52) Using (5.35), (5.52) and Hölder's inequality, we obtain some constant Cp5, s 0 , T q such that

E ˇˇg s 1 τ ´gs τ ˇˇ5 ď 2 5´1 ¨E`| O s 1 τ ´Os τ | 5 ¨ˇż s 1 0 rO  τ s ´1R V,α  pτ q d ˇˇ5 ˘`E ˇˇˇˇż s 1 s rO  τ s ´1R V,α  pτ q d ˇˇˇˇ5 ' ď Cp5, s 0 , T q|s 1 ´s| 5 . (5.53) Recall that each s satisfies the SDE s τ pwq " s 0 `ż τ 0 m ÿ i"1
Hp s τ 1 pwq, e i q ˝dα s;i τ 1 pwq, @τ P r0, T s.

As before, we can treat it as a Euclidean SDE. Hence,

E ˇˇ s 1 τ ´ s τ ˇˇ5 ď 3 5´1 ˜ˇ s 1 0 ´ s 0 ˇˇ5 `Eˇˇż τ 0 m ÿ i"1 Hp s τ 1 , e i q ˝d`α s 1 ,i τ 1 ´αs,i τ 1 ˘ˇ5 `Eˇˇż τ 0 m ÿ i"1
`Hp s 1 τ 1 , e i q ´Hp s τ 1 , e i q ˘˝dα s 1 ,i τ 1

ˇˇ5

":

3 5´1 `piq `pjq `pkq ˘.
Clearly, piq ď C|s 1 ´s| 5 for some C depending on sup |V| and r g. For pjq, we have

pjq ď E ˇˇż τ 0 m ÿ i"1 Hp s τ 1 , e i q ˝pO s 1 τ 1 ´Os τ 1 qdB τ 1 q i ˇˇ5 `Eˇˇż τ 0 m ÿ i"1 Hp s τ 1 , e i qpg s 1 τ 1 ´gs τ 1 q i dτ 1 ˇˇ5 ": pjq 1 `pjq 2 .
where we use the superscript i to denote the i-th component of a vector. For pjq 1 , we can transfer the integral into Itô's form. Note that all Hp¨, e i q are C 1 vector fields on O r g p Ă M q with bounded first order differentials. Hence, using Lemma 4.7, Hölder's inequality and (5.52), we can conclude that there is some constant Cp5, T q such that

pjq 1 ď Cp5, T q `E`ż τ 0 |O s 1 τ 1 ´Os τ 1 | 2 dτ 1 ˘5 2 `E`ż τ 0 |O s 1 τ 1 ´Os τ 1 | 2 dτ 1 ˘5ď Cp5, T qpT 1 2 5´1 `T 5´1 q ż τ 0 `E|O s 1 τ 1 ´Os τ 1 | 5 `E|O s 1 τ 1 ´Os τ 1 | 25 ˘dτ 1
ď Cp5, T qp2s 0 q 5 |s 1 ´s| 5 .

For pjq 2 , we can use Hölder's inequality and (5.53) to conclude that pjq 2 ď Cp5, s 0 , T qT 5 |s 1 ´s| 5 .

For pkq, the same argument as for pjq gives some constant Cp5, s 0 , T q such that pkq ď Cp5, s 0 , T q

ż τ 0 E ˇˇ s 1 τ 1 ´ s τ 1 ˇˇ5 dτ 1 .
Altogether, there is some constant Cp5, s 0 , T q such that

E ˇˇ s 1 τ ´ s τ ˇˇ5 ď Cp5, s 0 , T q ˆ|s 1 ´s| 5 `ż τ 0 E ˇˇ s 1 τ 1 ´ s τ 1 ˇˇ5 dτ 1 ˙.
The inequality also holds for sup

r τ Pr0,τ s E ˇˇ s 1 r τ ´ s r τ
ˇˇ5 . Hence we can apply Lemma 5.16 to conclude that there is some constant Cp5, s 0 , T q such that (5.54)

E ˇˇ s 1 τ ´ s τ ˇˇ5 ď Cp5, s 0 , T q|s 1 ´s| 5 .
This finishes the proof of (5.50) and hence (5.49) holds true. By Lemma 5.15, we can obtain a continuous modification of Υ V,α s ptq in the parameter pt, sq.

Let α s , s be as above. By (5.54) and Lemma 5.15, s has a version such that s Þ Ñ s pwq is continuous. By Lemma 5.11, to show s is differentiable in s, it suffices to show (5.36) is uniquely solvable with Y s 0 " p s 0 q 1 s . Let Y s,0 " 0 and let Y s,n (n ě 1) be such that " dθpY s,n t q " `Y s,n´1 t ˘˝dα s t `˝dΥ V,α s , d pY s,n t q " p s t q ´1R ` s t ˝dα s t , s t θpY s,n´1 t q ˘ s t .

(5.55)

For a R m ˆFpR m q valued process pv, Qq tPr0,T s , let }pv, Qq} :"

a }v} 2 `}Q} 2 , where }v} 2 " E sup tPr0,T s |v t | 2 , }Q} 2 " E sup tPr0,T s |Q t | 2 .
We show the sequence pθ, qpY s,n q converges in norm } ¨}. Clearly,

(5.56) › › pθ, q `Y s,1 ˘´pθ, q `Y s,0 ˘› › ď CT }α s }. We continue to estimate }pθ, q `Y s,n ˘´pθ, q `Y s,n´1 ˘}, n ě 2. By (5.55), " d `θpY s,n t q ´θpY s,n´1 t q ˘" ` pY s,n t q ´ pY s,n´1 t q ˘˝dα s t , d ` pY s,n t q ´θpY s,n´1 t q ˘" p s t q ´1R ` s t ˝dα s t , s t `θpY s,n t q ´θpY s,n´1 t q ˘˘ s t . Following the above discussion on › › ,n´1 ´ ,n´2 › › , we can use Doob's inequality of submartingale and Lemma 4.7 to conclude that E sup tPr0, r ts ˇˇpθ, q `Y s,n ˘´pθ, q `Y s,n´1 ˘ˇ2 ď Cps 0 , T q ş r t 0 E sup tPr0,τ s ˇˇpθ, q `Y s,n´1 ˘´pθ, q `Y s,n´2 ˘ˇ2 dτ.

(5.57)

Iterating this inequality for n steps, which, together with (5.56), imply

E sup tPr0, r ts ˇˇpθ, q `Y s,n ˘´pθ, q `Y s,n´1 ˘ˇ2 ď 1 n! pCT `Cps 0 , T qq n r t n .
In particular, when r t " T , this is

› › pθ, q `Y s,n ˘´pθ, q `Y s,n´1 ˘› › ď 1 n! pCT `Cps 0 , T qq n T n .
Hence pθ, qpY s,n q converges in } ¨} with some limit pθ, qpY s q which solves (5.36).

Such a solution Y s is unique. Assume Y s is another solution to (5.36) with Y s 0 " p s 0 q 1 s . Then the same argument as for (5.57) shows that E sup tPr0, r ts ˇˇpθ, qpY s t q ´pθ, qpY s t q ˇˇ2 ď Cps 0 , T q ż r t 0 E sup tPr0,τ s ˇˇpθ, qpY s t q ´pθ, qpY s t q ˇˇ2 dτ, from which we can conclude Y s " Y s by Gronwall's Lemma.

By Corollary 5.12, the solution Y s to (5.36) is actually given by (5.39). Hence, to show the process Y s t has a continuous modification in the parameter pt, sq, it suffices to show both s t and pY s t q have a pt, sq-continuous version. Let 5 ą 4, t, t 1 P r0, T s with t ă t 1 and s, s 1 P r´s 0 , s 0 s. Using (5.54) and applying Burkholder's inequality and Hölder's inequality to the difference s t 1 ´ s t , we obtain

2 1´5 E ˇˇ s 1 t 1 ´ s t ˇˇ5 ď E ˇˇ s 1 t 1 ´ s t 1 ˇˇ5 `Eˇˇ s t 1 ´ s t
ˇˇ5 ď Cp5, s 0 , T q `|s 1 ´s| 5 `|t 1 ´t| 1 2 5 ˘. So Lemma 5.15 applies and shows that there is a version of s t which is continuous in the parameter pt, sq. Since pY s 0 q " 0, by (5.39),

pY s t q " ż t 0 p s τ q ´1R ` s τ dα s τ , spτ q s τ p s 0 q ´1V pF s yq ˘ s τ `ż t 0 p s τ q ´1p∇p s τ e i qRq ` s τ e i , spτ q s τ p s 0 q ´1V pF s yq ˘ s τ dτ.
Again, by Burkholder's inequality and Hölder's inequality, it is easy to deduce that

E ˇˇ pY s 1 t 1 q ´ pY s t 1 q ˇˇ5 ď Cps 0 , b, T q `Eˇˇż t 1 0 |α s 1 τ ´αs τ | 2 dτ ˇˇ5 2 `Eˇˇż t 1 0 | s 1 τ ´ s τ | 2 dτ ˇˇ5 2 `E ż t 1 0 | s 1 τ ´ s τ | 5 dτ ď Cps 0 , b, T q ż t 1 0 `E|α s 1 τ ´αs τ | 5 `E| s 1 τ ´ s τ | 5 ˘dτ ď Cp5, s 0 , T q|s 1 ´s| 5 and E ˇˇ pY s t 1 q ´ pY s t q ˇˇ5 ď Cps 0 , b, T q ´|t 1 ´t| 5 2 `|t 1 ´t| 5 ¯ď Cps 0 , b, T q|t 1 ´t| 5 2 . Hence 2 1´5 E ˇˇ pY s 1 t 1 q ´ pY s t q ˇˇ5 ď E ˇˇ pY s 1 t 1 q ´ pY s t 1 q ˇˇ5 `Eˇˇ pY s t 1 q ´ pY s t q ˇˇ5 ď Cp5, s 0 , T q `|s 1 ´s| 5 `|t 1 ´t| 1 2 5 ˘,
which implies that pY s t q has a pt, sq-continuous modification by Lemma 5.15. Now we have shown i) and ii). Hence we can use Lemma 5.11 to conclude that y s " I s 0 pα s q is differentiable in s and satisfies (5.45). The differential process Υ V,y s pt, wq " sptq s t p s 0 q ´1V pF s yq has a pt, sq-continuous version since s t does. Finally, by Lemma 5.14, for iv), it suffices to show for v s t 1 P T s t 1

Fp Ă M q C 1 in s, (5.43) is uniquely solvable with initial pv s t 1 q 1 s . Again, this can be done by Picard's iteration method. Let υ s,0 t " pθ, q ´1 s t pθ, q s t 1 pv s t 1 q 1 s . For n ě 1, let υ s,n t with initial pv s t 1 q 1 s be such that

$ ' ' & ' ' % d `θpυ s,n t q
˘" `υs,n´1

where v s t , f I `vs t , p s t q 1 s ˘, f θ A `vs t , p s t q 1 s ˘, f A
`vs t , p s t q 1 s ˘are as in Lemma 5.14. We will show pθ, qpυ s,n t q converges in norm }¨}, where, for any R m ˆFpR m q valued process pv, Qq tPrt 1 ,t 2 s , }pv, Qq} :"

a }v} 2 `}Q} 2 , }v} 2 " E sup tPrt 1 ,t 2 s |v t | 2 and }Q} 2 " E sup tPrt 1 ,t 2 s |Q t | 2 .
Clearly, we have

› › pθ, qpυ s,1 q ´pθ, qpυ s,0 q › › ă Cps 0 , T q `› › › › ż t t 1 θ `fI `vs τ , p s τ q 1 s ˘˘› › › › `› › › › ż t t 1 `fI `vs τ , p s τ q 1 s ˘˘› › › › `› › › › ż t t 1 f θ A `vs τ , p s τ q 1 s ˘› › › › `› › › › ż t t 1 f A `vs τ , p s τ q 1 s ˘› › › › ": Cps 0 , T q `pAq 1 `pAq 2 `pAq 3 `pAq 4 .
Using Doob's inequality of submartingale, Lemma 4.7 and Hölder's inequality, we see from the expressions of f I `vs t , p s t q 1 s ˘, f θ A `vs t , p s t q 1 s ˘and f A `vs t , p s t q 1 s ˘that (5.58) So, using Doob's inequality of sub-martingales and Lemma 4.7, we compute that

pAq i ď Cps 0 , T q `}pθ, qpv s t q} `› › |pθ, qpv s t q| 2 › › ¨› › |p s t q 1 s | 2 › › ˘, i " 1, 2,
E sup tPrt 1 , r ts ˇˇpθ, qpv s t q ˇˇ5 ď Cps 0 , T q ż r t t 1 E sup tPrt 1 ,τ s ˇˇpθ, qpv s t q ˇˇ5 dτ, 5 " 2, 4,
which, by Gronwall's lemma, implies

(5.59) }pθ, qpv s t q}, › › |pθ, qpv s t q| 2 › › ď Cps 0 , T q. With a similar computation, we conclude from (5.39) that }|p s t q 1 s | 2 } is also bounded by constant Cps 0 , T q. So, (5.60) › › pθ, qpυ s,1 q ´pθ, qpυ s,0 q › › ď Cps 0 , T q.

For n ě 2, the difference pθ, q `υs,n ˘´pθ, q `υs,n´1 ˘satisfies the SDE As before, we can use Doob's inequality of sub-martingales and Lemma 4.7 to obtain E sup tPrt 1 , r ts ˇˇpθ, q `υs,n t ˘´pθ, q `υs,n´1 t ˘ˇ2 ď Cps 0 , T q ş r t

$ ' ' ' & ' ' ' % d `θpυ s,n t q ´θpυ s,n´1 t q ˘" ` pυ s,
t 1 E sup tPr0,τ s ˇˇpθ, q `υs,n´1 t ˘´pθ, q `υs,n´2 t ˘ˇ2 dτ. (5.61) Iterate this inequality for n steps and then let r t " t 2 . This, together with (5.60), implies › › pθ, q `υs,n ˘´pθ, q `υs,n´1 ˘› › ď

1 n! Cps 0 , T q n T n .
Hence pθ, qpυ s,n q converges in } ¨} with some limit pθ, qpυ s q which solves (5.43). We can also use (5.61) and Gronwall's Lemma to conclude the uniqueness of such υ s t . For the existence of a continuous version of υ s t in the pt, sq parameter, we use Lemma 5.15. Let 5 ą 4, t, t 1 P rt 1 , T s with t ă t 1 and s, s 1 P r´s 0 , s 0 s. Using (5.58) and Lemma 4.7, we deduce that

E ˇˇpθ, qpυ s 1 t 1 q ´pθ, qpυ s t 1 q ˇˇ5 ď Cp5, s 0 , T q `|s 1 ´s| 5 `ż t 1 t 1 E ˇˇpθ, qpυ s 1 τ q ´pθ, qpυ s τ q ˇˇ5 dτ ˘,
which, by Gronwall's lemma, implies E ˇˇpθ, qpυ s 1 t 1 q ´pθ, qpυ s t 1 q ˇˇ5 ď Cp5, s 0 , T q|s 1 ´s| 5 .

Similarly, it is true that

E ˇˇpθ, qpυ s t 1 q ´pθ, qpυ s t q ˇˇ5 ď Cp5, s 0 , T qp › › |pθ, qυ s | 1 2 5 › › `› › |pθ, qv| 1 2 5 › › `› › |pθ, qv| 1 2 5 › › ¨› › |p s q 1 s | 1 2 5 › › q `|t 1 ´t| 1 2 5 `|t 1 ´t| 5 ď Cp5, s 0 , T q|t 1 ´t| 1 2 5 ,
where, to obtain the last inequality, we first show }|p s q 1 s | 1 2 5 } ă Cp5, s 0 , T q by (5.39) and then argue as for (5.59) to show }|v|

1 2 5 }, › › |υ s | 1 2 5 ›
› is also bounded by some Cp5, s 0 , T q. Thus,

2 1´5 E ˇˇpθ, qpυ s 1 t 1 q ´pθ, qpυ s t q ˇˇ5 ďE ˇˇpθ, qpυ s 1 t 1 q ´pθ, qpυ s t 1 q ˇˇ`E ˇˇpθ, qpυ s 1 t 1 q ´pθ, qpυ s t 1 q ˇˇ5 ďCp5, s 0 , T q ´|s 1 ´s| 5 `|t 1 ´t| 1 2 5 ¯.
By Lemma 5.15, there is a continuous modification of pθ, qpυ s t q in the pt, sq parameter. Note that pθ, q s t varies continuously with respect to s t , which is also continuous in the pt, sq parameter. So, we can also obtain a pt, sq-continuous version of the process υ s t .

Remark 5.18. As we will see in the proof of Proposition 5.19, for almost all w, α s 1 ˝αs 2 pwq " α s 1 `s2 pwq, for all s 1 , s 2 P R.

Hence, intuitively, tα s u sPR introduced a one parameter family of 'flow' maps on Brownian paths starting from the o P R m . Consequently, tF s u sPR also behaves like a one parameter family of 'flow' maps which satisfy the cocycle property F s 1 ˝Fs 2 " F s 1 `s2 for any s 1 , s 2 P R.

5.4. Quasi-invariance property of F s y . Let y s " F s y y be as in Theorem 5.17. We continue to study its distribution using the classical Cameron-Martin-Girsanov formula.

Let py t , t q be the stochastic process pair which defines the Brownian motion on p Ă M , r gq starting from y up to time T , i.e., y t " πp t q and t P O r g p Ă M q solves the Stratonovich SDE

d t " m ÿ i"1
Hp t , e i q ˝dB i t pwq, @t P r0, T s.

By an abuse of notation, we continue to use P y to denote the Brownian distribution in C y pr0, T s, Ă M q (i.e., the distribution of py t q tPr0,T s ) and use Q to denote the distribution of pB t q tPr0,T s in C o pr0, T s, R m q. Using the Itô map, we have the relation B " pI 0 q ´1pyq and P y " Q ˝pI 0 q ´1. Similarly, let P F s y denote the Brownian motion distribution on C F s y pr0, T s, Ă M q. Then

P F s y " Q ˝pI s 0 q ´1.
Let y s and α s be the one parameter family of stochastic processes on Ă M and in R m , respectively, that we obtained in Theorem 5.17. They are related by the identity α s " pI s 0 q ´1py s q. Let P s , Q s be the distributions of y s , α s , respectively, where Q 0 " Q. Then P s " Q s ˝pI s 0 q ´1. (5.62) To compare P s with P F s y , it suffices to compare Q s with Q 0 , which can be understood by a simple application of the Cameron-Martin-Girsanov formula.

Proposition 5.19. The distribution Q s is equivalent to Q 0 with dQ s dQ 0 pwq " e t 1 2 ş T 0 xg s t pα ´spwqq, dBtpwqy´1 4 ş T 0 |g s t pα ´spwqq| 2 dtu . (5.63)
Consequently, the distribution P s is equivalent to the Brownian distribution P F s y with (5.64) dP s dP F s y pβq " dQ s dQ 0 `pI s 0 q ´1pβq ˘, β P C F s y pr0, T s, Ă M q.

Proof. We follow the proof of [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]Theorem 3.5]. Clearly, (5.64) follows from (5.63) by using the identity (5.62). For (5.63), recall that Q s is the distribution of pα s t q tPr0,T s , where (Because of (5.65), the composition α s 1 ˝αs 2 , s 1 , s 2 P R, is well-defined and has a continuous version in the parameter ps 1 , s 2 q using Kolmogorov's criterion as in Theorem 5.17. So, by the uniqueness of the α s family and its continuous in s, we must have α s 1 ˝αs 2 " α s 1 `s2 . In particular, (5.67) holds true.) Now, from (5.67), we deduce Proposition 5.20. The probability P F s y ˝Fs y is absolutely continuous with respect to P y and the Radon-Nikodyn derivative dP F s y ˝Fs y {dP y conditioned on y T " x is L q integrable for every q ě 1, locally uniformly in the s parameter. Moreover, dP F s y ˝Fs y {dP y is differentiable in s with differential E s T pdP F s y ˝Fs y {dP y q, where E s T conditioned on y T " x is also L q integrable for every q ě 1, locally uniformly in the s parameter.

α
O
Proof. For P y almost all path β, let w " I ´1 0 pβq. Then I ´1

ż s 0 rO  τ s ´1 p  0 q ´1 " s 1 pτ qVpF  yq ‰ ´Ric `  τ p  0 q ´1rspτ qVpF  yqs ˘( d.
Put E s t pwq :" e t´1 2 ş t 0 xg s τ pwq, dBτ pwqy`1 4 ş t 0 |g s τ pwq| 2 dτ u , @t P r0, T s. For q ě 1, we estimate E P ẙ,x,T |E s T pwq| q . Let b t be the Brownian motion with respect to the bridge distribution (from y to x in time T ) as in Lemma 4.15 such that dB τ pwq " db τ pwq `2 ´1 τ ∇ ln ppT ´τ, y τ , xq dτ.

Then conditioned on y T " x, |E s T pwq| q has the same distribution as e t´1 2 q ş T 0 xg s τ pwq, dbτ pwqy`1 4 q

ş T 0 |g s τ pwq| 2 dτ ´q ş T 0 xg s τ pwq, ´1 τ ∇ ln ppT ´τ,yτ ,xqy dτ u .
So, by Hölder's inequality and the Cameron-Martin-Girsanov Theorem,

E P ẙ,x,T |E s T pwq| q ď a ppT, x, yq " E P y,x,T e t´ş T 0 qxg s τ pwq, dbτ pwqy´q 2 ş T 0 |g s τ pwq| 2 dτ u ı 1 2 ¨"E P ẙ,x,T e t´2q ş T 0 xg s τ , ´1 τ ∇ ln ppT ´τ,yτ ,xqy dτ `p 1 2 q`q 2 q ş T 0 |g s τ | 2 dτ u ı 1 2 ď a ppT, x, yq " E P ẙ,x,T e t´2q ş T 0 xg s τ , ´1 τ ∇ ln ppT ´τ,yτ ,xqy dτ `p 1 2 q`q 2 q ş T 0 |g s τ | 2 dτ u ı 1 2 . (5.68)
Let us continue to use C to denote a constant depending on }g} C 3 , m and the norm bound of V and use Cp¨q to indicate the extra coefficients it depends on. By our choice of s (see (5.24)), for s P r´s 0 , s 0 s, |g s τ pwq| ď Cps 0 , T q for some Cps 0 , T q. Apply this in (5.68) and then use (4.32) and (4.33). We obtain some Cpq, s 0 , T q, r Cpq, s 0 , T q such that E P ẙ,x,T |E s T pwq| q ď Cpq, s 0 , T q

" E P ẙ,x,T e t r
Cpq,s 0 ,T q ş T 0 }∇ ln ppT ´τ,yτ ,xq} dτ ı 1 2 , which, by (4.34), shows that dP F s y ˝Fs {dP y conditioned on y T " x is L q integrable for q ě 1, locally uniformly in the s parameter. Hence the Radon-Nikodyn derivative dP F s y ˝Fs y {dP y is differentiable in s with differential pE s T q 1 s pwq, which, by using stochastic Duhamel principle (or Itô's formula), is

pE s T q 1 s " E s T ¨ˆpE s 0 q 1 s ´1 2 ż T 0 xpg s t q 1 s pwq, dB τ pwqy `1 2 ż T 0 xpg s t q 1 s pwq, g s t pwqy dτ " E s T ¨ˆ´1 2 ż T 0 xpg s t q 1 s pwq, dB t pwqy `1 2 ż T 0 xpg s t q 1
s pwq, g s t pwqy dt ":

E s T ¨Es T .
Conditioned on y T " x, E s T has the same distribution as

´1 2 ż T 0 xpg s τ q 1 s , db τ y `1 2 ż T 0 xpg s τ q 1 s , g s τ y dτ ´ż T 0 @ pg s τ q 1 s , ´1 τ ∇ ln ppT ´τ, y τ , xq D dτ,
where both |g s t | and |pg s t q 1 s | are bounded by some constant Cps 0 , T q. Hence, by Hölder's inequality and (4.22), we compute that

`EP y,x,T |E s T | q ˘2 ď 3 2q´1 ˜EP y,x,T ˇˇˇż T 0 xpg s τ q 1 s , db τ y ˇˇˇ2 q `pCps 0 , T qq 2q `EP y,x,T ˇˇˇż T 0 @ pg s τ q 1 s , ´1 τ ∇ ln ppT ´τ, y τ , xq D dτ ˇˇˇ2 q ": 3 2q´1 ´pIq `pIIq `pIIIq ¯.
Since b is a Brownian motion with respect to P y,x,T , by Lemma 4.7,

pIq ď C 1 p2qq ż T 0 E P y,x,T |pg s τ q 1 s | 2q dτ ď Cpq, s 0 , T q,
where C 1 is from (4.21). Using |pg s τ q 1 s wq| ď Cps 0 , T q and (4.34), we obtain pIIIq ďCps 0 , T q 2q E P y,x,T `ż T 0 }∇ ln ppT ´τ, y τ , xq} dτ ˘2q ďCps 0 , T q 2q E P y,x,T e t2q ş T 0 }∇ ln ppT ´τ,yτ ,xq} dτ u ďCps 0 , T q 2q pppT, x, yqq ´1e c 1 p1`dpx,yqq .

Putting all the estimations on pIq, pIIq and pIIIq together, we conclude that E s T conditioned on y T " x is L q integrable for q ě 1, locally uniformly in the s parameter.

Consider the distribution of P x on C x pr0, T s, Ă M q. Let px, uq be the stochastic pair which defines the Brownian motion on p Ă M , r gq which starts from x. The distribution of px t q tPr0,T s is independent of the choice of u 0 . Hence P px,u 0 q , which is the distribution of px t q tPr0,T s with a initial frame u 0 , coincides with P x on C x pr0, T s, Ă M q and P px,u 0 q,y,T :" E P px,u 0 q `¨ˇˇx T " y coincides with P x,y,T on C x,y pr0, T s, Ă M q. This means

P x "
ż ż P px,u 0 q,y,T ¨ppT, x, yq dVolpyq dVolpu 0 q

ˆ" ż ż P x,y,T ¨ppT, x, yq dVolpu 0 q dVolpyq " ż ż P px,u 0 q,y,T ¨ppT, x, yq dVolpu 0 q dVolpyq, where dVolpu 0 q is the uniform distribution on O r g x p Ă M q. For any y P Ă M , the Brownian bridge process connecting x and y in time T has the following symmetric property.

Lemma 5.21. Let pX t , U t q tPr0,T s be the pair of stochastic processes for Brownian bridge from x to y in time T . i) Under P x,y,T , the process pX T ´tq tPr0,T s has the law P y,x,T . ii)

If U 0 is chosen randomly with the uniform distribution in O r g x p Ă M q, then U T is also uniformly distributed in O r g y p Ă M q.
Proof. i) is [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]Proposition 5.4.3]. (It is true since by (4.36), the finite margin of X, or the joint density function of X t 1 , ¨¨¨, X tn , 0 " t 0 ă t 1 ă ¨¨¨ă t n ă t n`1 " T , is given by 1 ppT, x, yq n ź i"0 ppt i`1 ´ti , x i , x i`1 q, where x 0 " x, x n`1 " y, which is the same as the joint density function of r X T ´tn , ¨¨¨, r X T ´t1 of the bridge r X from y to x in time T .) For ii), we consider (4.37). Note that the distribution of the R m -Brownian motion b t is invariant under rotations. So if pU t q tPr0,T s solves (4.37) with initial frame U 0 , then for r U 0 " U 0 υ with υ P OpR m q, p r U t " U t υq tPr0,T s solves (4.37). This implies ii).

Let F s y be as in Theorem 5.17. It induces a map from C y,x pr0, T s, Ă M q to C F s y,x pr0, T s, Ă M q. We define F s on C x pr0, T s, Ă M q conditioned on the value of β T by letting

F s pβq :" F s β T pβq.
By Lemma 5.21, a uniform random choice of u 0 at x will result in a uniform distribution of u T at y for the Brownian bridge connecting x and y in time T . Therefore, to analyze P x ˝Fs , we can choose the initial 0 P O r g β T p Ă M q with a uniform distribution to define F s β T .

Lemma 5.22. For P x almost all β P C x pr0, T s, Ă M q,

(5.69)

dP x ˝Fs dP x pβq " dP F s β T ˝Fs dP β T pβq ¨dVolpF s β T q dVolpβ T q .
Proof. Lemma 5.21 implies that the distribution of u T is uniform if u 0 is. So if we disintegrate P x according to the value of px T , u T q, we obtain

P x "
ż ż P px,u 0 q,y,T ¨ppT, x, yqdVolpu 0 qdVolpyq " ż ż P x,py, 0 q,T ¨ppT, x, yqdVolp 0 qdVolpyq, where dVolp 0 q is the uniform probability on O r g y p Ă M q. For any measurable subset A Ă C x pr0, T s, Ă M q, by the change of variable formula, P x pF s pAqq " ż ż P x,pF s y,F s 0 q,T pF s pAqq ¨ppT, x, F s yq dVolpF s 0 q dVolpF s yq " ż ż P x,pF s y,F s 0 q,T pF s pAqq ¨ppT, x, F s yq dVol ˝F s dVol pyq dVolp 0 q dVolpyq. By Lemma 5.21, the distribution of P x,pF s y,F s 0 q,T on C x,F s y pr0, T s, Ă M q " C F s y,x pr0, T s, Ă M q can be identified with that of P pF s y,F s 0 q,x,T , the Brownian bridge from F s y to x in time T with the initial frame F s 0 P O r g F s y p Ă M q. Hence (5.70) P x pF s pAqq " ż ż P pF s y,F s 0 q,x,T pF s pAqq ¨ppT, F s y, xq dVol ˝F s dVol pyq dVolp 0 q dVolpyq.

The absolute continuity of P x ˝Fs with respect to P x will follow if P pF s y,F s 0 q,x,T ˝Fs is absolutely continuous with respect to P py, 0 q,x,T and the Radon-Nikodym derivative dP pF s y,F s 0 q,x,T ˝Fs {dP py, 0 q,x,T is integrable. Since the bridge process from y to x in time T is just the conditional process of y on y T " x, Lemma 5.19 implies that P pF s y,F s 0 q,x,T ˝Fs is absolutely continuous with respect to P py, 0 q,x,T .

As to (5.69), we see that for any measurable set r A Ă C y pr0, T s, Ă M q, ˝Fs {dP x conditioned on x T " y is L q integrable for every q ě 1, locally uniformly in the s parameter. The differential of dP x ˝Fs {dP x in s exists and is of the form Ð Ý E s T ¨pdP x ˝Fs {dP x q, where Ð Ý E s T conditioned on x T " y is square integrable, locally uniformly in the s parameter.

Using (5.69) and the proof of Proposition 5.20, we can deduce that Ð Ý E s T differs from E s T by the differential of dVolpF s yq{dVolpyq in the s parameter, where E s T can be understood as a backward stochastic integral on the bridge paths from x to y in time T . 5.5. The extended map F s . In order to show the properties iii), iv) of F s in Section 5.1, we need to clarify pDπptu T s λ q p1q λ q ˝Fs for Φ 1 λ ˝Fs , where Φ 1 λ is as in (5.14). We will achieve this by extending F s to the process ptu T s λ q p1q λ and letting ´Dπptu T s λ q p1q λ ¯˝F s :" Dπ ´ptu T s λ q p1q λ ˝Fs ¯.

The rough idea is that the maps F s on orbits extend naturally to their tangent maps for the parallel transportations and hence can be defined for the objects they make.

We first deal with ptu T s λ q

p1q 0 ˝Fs . Let λ Þ Ñ tu 0 s λ P O r g λ p Ă M q be C k´2 in Fp Ă
M q and let ptu t s λ P O r g λ p Ă M qq tPr0,T s with initials tu 0 s λ be the unique solution to (5.73)

dtu t s λ " m ÿ i"1
H λ ptu t s λ , e i q ˝dB i t pwq, @t P r0, T s.

By Lemma 4.17, there is a version of ttu t s λ u such that λ Þ Ñ tu t s λ pwq is C k´2 in λ for almost all w. By Lemma 4.20, the differential process ptu t s λ q p1q 0 is given by (5.74)

ptu T s λ q p1q 0 " " D Ý Ñ F 0,T pu 0 , wq ı ptu 0 s λ q p1q 0 `ż T 0 " D Ý Ñ F t,T pu t , wq ı pH λ q p1q 0 pu t , e i q ˝dB i t pwq,
where u " tus 0 and tD Ý Ñ F t,t u 0ďtătďT are the tangent maps of the flow maps t Ý Ñ F t,t u 0ďtătďT associated to (5.73) at λ " 0 (the arrow is to indicate the time is recorded starting from x). By Lemma 4.4 (see also Lemma 5.13), the tD Ý Ñ F t,T u are determined by the paths px τ pwq " πpu t pwqqq τ Pr0,T s (or its anti-development in R m ). Hence (5.74) shows that ptu T s λ q p1q 0 pwq are objects completely determined by px τ pwqq τ Pr0,T s , ptu 0 s λ q p1q 0 and pH λ q p1q 0 .

By symmetry of the Brownian motion, we can describe the distribution of ptu T s λ q p1q 0 conditioned on x T " y using py t , t q tPr0,T s , which is the stochastic pair defining the Brownian motion on p Ă M , r gq starting from y. The two path spaces C y,x pr0, T s, Ă M q and C x,y pr0, T s, Ă M q can be identified. Moreover, the distribution of y conditioned on y T " x coincides with x conditioned on x T " y. This means for almost all such path py τ q τ Pr0,T s pwq ": β, it is associated with a path px t q tPr0,T s pwq " pβ T ´τ q τ Pr0,T s ": Ý Ñ β . So the stochastic parallel transportation of u t along Ý Ñ β is well-defined and is given by u t " T ´tp T q ´1u 0 .

For any element X P T ut Fp Ă M q, let pθ, q ´1 ut X ": pX 1 , X 2 q.

Note that the orthonormal frames u t and T ´t have the same footpoint x t pωq " y T ´tpwq.

Hence X also naturally corresponds to an element YpXq " Y in T T ´t Fp Ă M q with pθ, q ´1 T ´t Y :" ` ´1 T ´tu t pX 1 q, Adp ´1 T ´tu t qpX 2 q ˘.

We see that X and YpXq are just the same vector expressed in different frame charts. Denote by Y this map which sends tangents X P T uτ Fp Ă M q to YpXq P T T ´τ Fp Ă M q for any τ P r0, T s. Let pF t 1 ,t 2 q 0ďt 1 ăt 2 ďT and pDF t 1 ,t 2 q 0ďt 1 ăt 2 ďT be the invertible stochastic flow maps and tangent maps associated to y (cf. (5.40)). The following is true.

Lemma 5.24. Let β, X, Y be introduced as above. Then for almost all β, we have

(5.75) Y `DÝ Ñ F t,T pu t , wqX ˘" DpF 0,T ´tp 0 , wqq ´1pYpXqq " " DF 0,T ´tp 0 , wq ‰ ´1pYpXqq.
Proof. By Corollary 4.2, for almost all w, the maps F 0,T ´tp¨, wq are C k´2 diffeomorphisms.

So for almost all w, the tangent maps DpF 0,T ´tp 0 , wqq ´1 and rDF 0,T ´tp 0 , wqs ´1 exist and are equal. For (5.75), it suffices to verify the first equality.

Write pθ, q ´1 ut X ": pX 1 t , X 2 t q and let pX 1 τ , X 2 τ q :" pθ, q ´1 uτ D Ý Ñ F t,τ pu t , wqX, @τ P rt, T s.

It is true by Lemma 5.13 that dX 1 τ " X 2 τ ˝dB τ pwq, (5.76)

dX 2 τ " pu τ q ´1R `uτ ˝dB τ pwq, u τ X 1 τ ˘uτ . (5.77) Let pY 1 τ , Y 2 τ q :" pθ, q ´1 T ´τ Y `pθ, q uτ pX 1 τ , X 2 τ q ˘.
Note that p T ´τ q ´1u τ " p T q ´1u 0 . So (5.76) gives

dY 1 τ " p T ´τ q ´1u τ dX 1 τ " p T ´τ q ´1u τ X 2 τ ˝dB τ pwq " ´p T ´τ q ´1u τ X 2 τ `p T ´τ q ´1u τ ˘´1 ˝p T ´τ q ´1u τ d Ý Ñ B T ´τ pwq " ´Y2 τ ˝dÝ Ñ B T ´τ pwq,
where ˝Ý Ñ B t pwq denote the backward Stratonovich integral. Similarly, using (5.77), we obtain

dY 2 τ " pp T ´τ q ´1u τ qdX 2 τ pp T ´τ q ´1u τ q ´1 " p T ´τ q ´1R `uτ ˝dB τ pwq, u τ X 1 τ ˘ T ´τ " ´p T ´τ q ´1R ´ T ´τ ˝dÝ Ñ B T ´τ pwq, T ´τ Y 1 τ ¯ T ´τ .
Altogether, we have

dY 1 τ " ´Y2 τ ˝dÝ Ñ B T ´τ pwq, dY 2 τ " ´p T ´τ q ´1R ´ T ´τ ˝dÝ Ñ B T ´τ pwq, T ´τ Y 1 τ ¯ T ´τ
and the solution pY 1 T , Y 2 T q is exactly pθ, q 0 `DpF 0,T ´tp 0 , wqq ´1pYpXqq ˘.

As a corollary of (5.74) and Lemma 5.24, we have Corollary 5.25. Conditioned on x T " y, the distribution of ptu T s λ q p1q 0 given by (5.74) is the same as, conditioned on y T " x, the distribution of ptu T s λ q p1q 0 :" rDF 0,T p 0 , wqs ´1ptu 0 s λ q p1q 0 ´ż T 0 rDF 0,t p 0 , wqs ´1pH λ q p1q 0 p t , e i q ˝dÝ Ñ B i t pwq, where ˝dÝ Ñ B t pwq is the backward Stratonovich infinitesimal.

Proof. Consider the mapping pH λ q p1q 0 pu t , ¨q " pH λ q p1q 0 p T ´tp T q ´1u 0 , ¨q from T o R m to T ut Fp Ă M q. We have pH λ q p1q 0 pu t , e i q ˝dB i t pwq " pH λ q p1q 0 pu t , ˝dB t pwqq and its correspondence at T T ´t Fp Ă M q is ´pH λ q p1q 0 p T ´t, ˝dÝ Ñ B T ´tpwqq. So, by Lemma 5.24,

" D Ý Ñ F t,T pu t , wq ı pH λ q p1q 0 pu t , ˝dB t pwqq " ´rDF 0,T ´tp 0 , wqs ´1pH λ q p1q 0 p T ´t, ˝dÝ Ñ B T ´tpwqq
and the conclusion follows by taking the integral with respect to t on r0, T s.

Let α s , y s and s be the processes obtained in Theorem 5.17. Let pF s t 1 ,t 2 q 0ďt 1 ăt 2 ďT be the parallel transportation stochastic flow of y s and let rDF s t 1 ,t 2 p s t 1 , wqs be the associated tangent maps. By Proposition 4.1, rDF s 0,t p s t , wqs is invertible for almost all w. Hence the inverse maps rDF s 0,t p s 0 , wqs ´1 are well-defined. Corollary 5.25 shows the distribution of ptu T s λ q p1q 0 pwq is the same as ptu T s λ q p1q 0 pwq. We define ptu T s λ q p1q 0 pwq ˝Fs :" ptu T s λ q p1q 0 pwq ˝Fs " ptu s T s λ q p1q 0 pwq, where ptu s T s λ q p1q 0 :" rDF s 0,T p s 0 , wqs ´1ptu 0 s λ q p1q 0 ´ż T 0 rDF s 0,t p s 0 , wqs ´1pH λ q p1q 0 p s t , e i q ˝dÝ Ñ α s,i t pwq.

So the differentiability of ptu T s λ q p1q 0 ˝Fs in s will follow from the differentiability of ptu s T s λ q p1q 0 in s, which is intuitively true by the differentiability of (in s) of α s t , s t and rDF s 0,t p s 0 , ¨qs ´1. We will justify this and formulate pptu T s λ q p1q 0 ˝Fs q 1 s in the remaining part of this subsection.

Lemma 5.26. Let α s t , O s t , g s t , Υ V,α  and s t be as in Theorem 5.17. Fix T 0 ą 0. For any s 0 ą 0, q ě 1 and T ą T 0 , there are constants c A (which depends on s 0 , m, q, s and }g 0 } C 3 ) and c A (which depends on m, q, T, T 0 and }g 0 } C 3 ) such that (5.78) sup sPr´s 0 ,s 0 s

E P ẙ,x,T sup tPr0,T s |A| q ă c A e c A p1`d r g λ px,yqq ,
where A " α s t , pO s t q 1 s , pg s t q 1 s , Υ V,α s , p s t q 1 s or pθ, q `p s t q 1 s ˘.

Proof. By our construction, α s t " ş t 0 O s τ dB τ `gs τ dτ , where O s P OpR m q and |g s | ď cs 0 sup |V| for some c that bounds sup tPr0,T s t|s|, |s 1 |u, supt}Ric}u. So,

2 1´q E P ẙ,x,T sup tPr0,T s |α s t | q ďE P ẙ,x,T sup tPr0,T s › › › › ż t 0 O s τ dB τ › › › › q `c0
T ´m 0 pcs 0 T sup |V|q q e c 0 p1`T q ":pIq `c0 T ´m 0 pcs 0 T sup |V|q q e c 0 p1`T q , where c 0 , c 0 are from (4.32). Let b be the Brownian motion in Lemma 4.15 for P y,x,T , i.e.,

(5.79) dB τ " db τ `2p 0 τ q ´1∇ ln ppT ´τ, y 0 τ , xq dτ.

Then,

pIq "

E P ẙ,x,T sup tPr0,T s › › › › ż t 0 O s τ db τ `2O s τ p 0 τ q ´1∇ ln ppT ´τ, y 0 τ , xq dτ › › › › q ď 2 q´1 E P ẙ,x,T sup tPr0,T s › › › › ż t 0 O s τ db τ › › › › q `22q´1 E P ẙ,x,T ˇˇˇż T 0 › › ∇ ln ppT ´t, y 0 t , xq › › dt ˇˇˇq ": pIq 1 `pIq 2 .
For pIq 1 , by successively using Doob's inequality of submartingale, Hölder's inequality and Burkholder's inequality, we obtain

2 1´q pIq 1 ď CpqqppT, x, yqE P y,x,T › › ż T 0 O s τ db τ › › q ď CpqqppT, x, yq ˆEP y,x,T › › ż T 0 O s τ db τ › › 2q ˙1 2 ď c 0 T ´m 0 CpqqC 1 pqq ? T q e c 0 p1`T q ,
where Cpqq " pq{q ´1q q and C 1 p¨q is as in Lemma 4.7. For pIq 2 , by Proposition 4.14, 2 1´2q pIq 2 ď E P ẙ,x,T ´eq ş T 0 }∇ ln ppT ´t,y 0 t ,xq} dt ¯ă e cp1`dpx,yqq , where c is as in (4.34). Putting the estimations together, we obtain (5.78) for A " α s t . Next, we consider (5.78) for pO s t q 1 s , pg s t q 1 s . By (5.34) and (5.35), we have pO s t q 1 s " ´KV,α s pτ qO s t , pg s t q 1 s " ´KV,α s pτ qg s t `p s 0 q ´1rs 1 ptqVpF s yqs ´Ric

` s t p s 0 q ´1rsptqVpF s yqs ˘,
where

K V,α s ptq " ż t 0 p s τ q ´1R ` s τ dα s τ , s τ p s 0 q ´1rspτ qVpF s yqs ˘ s τ `ż t 0 p s τ q ´1p∇p s τ e i qRq
` s τ e i , s τ p s 0 q ´1rspτ qVpF s yqs ˘ s τ dτ.

Since O s P OpR m q, |g s | ď cs 0 sup |V|, and all the |s|, |s 1 | and |V| are uniformly bounded, it is clear that (5.78) holds for pO s t q 1 s , pg s t q 1 s if it holds for K V,α s ptq. Using (5.79) and (4.22),

we obtain

E P ẙ,x,T sup tPr0,T s |K V,α s ptq| q ď 3 q´1 E P ẙ,x,T sup tPr0,T s ˇˇˇż t 0 p s τ q ´1R ` s τ O s τ db s τ , s τ p s 0 q ´1rspτ qVpF s yqs ˘ s τ ˇˇˇq `3q´1 p2 sup }R} sup |Vq q E P ˚y,x,T ˇˇˇż T 0 }∇ ln ppT ´t, y 0 t , xq} dt ˇˇˇq `3q´1 psup }∇R} sup |V|s 0 T q q ,
which has the same bound type as in (5.78) by a computation similar to the one for pIq.

To verify (5.78) for Υ V,α s , it suffices to check it for K t :" ş t 0 xK V,α s pτ q, dα s τ y since Υ V,α s ptq " ż t 0 p s 0 q ´1rs 1 pτ qVpF s yqs ´Ric ` s τ p s 0 q ´1rspτ qVpF s yqs ˘´K t .

By (5.79) and (4.22),

3 1´q E P ẙ,x,T sup tPr0,T s |K t | q ď E P ẙ,x,T sup tPr0,T s ˇˇˇż t 0 xK V,α s pτ q, O s τ db τ y ˇˇˇq `EP ẙ,x,T sup tPr0,T s ˇˇˇż t 0 xK V,α s pτ q, 2 ´1 τ ∇ ln ppT ´τ, y 0 τ , xq dτ y ˇˇˇq `EP ẙ,x,T sup tPr0,T s ˇˇˇż t 0
xK V,α s pτ q, g s τ dτ y ˇˇˇq ": pIIq 1 `pIIq 2 `pIIq 3 .

For pIIq 1 , it is routine to apply successively Hölder's inequality, Doob's inequality of submartingale and Burkholder's inequality, which gives

ppIIq 1 q 2 ďppT, x, yqE P ẙ,x,T sup tPr0,T s ˇˇˇż t 0 xK V,α s pτ q, O s τ db τ y ˇˇˇ2 q ďCp2qqppT, x, yqE P y,x,T ˇˇˇż T 0 xK V,α s pτ q, O s τ db τ y ˇˇˇ2 q ďCp2qqC 1 p2qqppT, x, yqE P y,x,T ˇˇˇż T 0 }xK V,α s pτ q, O s τ y} 2 dτ ˇˇˇq ďCp2qqC 1 p2qqT q E P ẙ,x,T sup τ Pr0,T s |K V,α s pτ q| 2q .
For pIIq 2 , it is true that

ppIIq 2 q 2 ď2 2q E P ẙ,x,T sup τ Pr0,T s |K V,α s pτ q| 2q ¨EP ẙ,x,T ˇˇˇż T 0 }∇ ln ppT ´τ, y 0 t , xq} dτ ˇˇˇ2 q .
For pIIq 3 , a routine calculation shows pIIq 3 ď pcs 0 sup |V|q q T q E P ẙ,x,T sup τ Pr0,T s |K V,α s pτ q| q .

Putting the estimations on pIIq 1 , pIIq 2 and pIIq 3 together, we conclude from Proposition 4.14 and the estimation for K V,α s that (5.78) also holds true for K t . This shows (5.78) for Υ V,α s .

Finally, to check (5.78) for p s t q 1 s , pθ, q `p s t q 1 s ˘, it suffices to consider the latter, which holds true by the above conclusion for K V,α s since, by (5.39), θpY s t q " sptqp s 0 q ´1V pF s yq, pY s t q " K V,α s ptq.

Lemma 5.27. Let α s be as in Theorem 5.17. .

Let T 0 ą 0. For any s 0 ą 0, q ě 1 and T ą T 0 , there are constants c F (which depends on s 0 , m, q, s and }g 0 } C 2 ) and c F (which depends on s 0 , m, q, s, T, T 0 and }g 0 } C 3 ) such that sup sPr´s 0 ,s 0 s

E P ẙ,x,T sup 0ďtătďT › › rDF s t,t p s t , wqs ´1› › q , sup 0ďtătďT › › ›r Č pDF s t,t qp s t , wqs ´1› › › q ă c F e c F p1`d r g λ px,yqq . (5.80)
Proof. For (5.80), it suffices to consider the second estimation. Let s P r´s 0 , s 0 s and t, t P r0, T s with t ă t. For pv 0 , Q 0 q P T o FpR m q, let pv t´τ , Q t´τ q :" r Č pDF s τ,t qp s 0 , wqs ´1pv 0 , Q 0 q, @τ P rt, ts. Then Lemma 5.13 shows that z τ :" pv τ , Q τ q satisfies the Itô form SDE dz t´τ pwq " m ÿ j"1 `´M j p s τ qz t´τ pwq d Ð Ý α s,j τ pwq `rM j p s τ qs 2 z t´τ pwq dτ ˘`Np s τ qz t´τ pwq dτ, where M j , N are given in (4.25), (4.26).The remaining estimation for (5.80) can be done by following the proof of Proposition 4.16.

Lemma 5.28. Let α s be as in Theorem 5.17. Then ppDF s 0,t q ´1q tPr0,T s , p Č pDF s 0,t q ´1q tPr0,T s are C 1 in the s parameter. Let T 0 ą 0. For any s 0 ą 0, q ě 1 and T ą T 0 , there exist c 1 F (which depends on s 0 , m, q, s and }g 0 } C 3 ) and c 1 F (which depends on s 0 , m, q, s, T, T 0 and

}g 0 } C 3 ) such that sup sPr´s 0 ,s 0 s E P ẙ,x,T sup tPr0,T s › › prpDF s 0,t qp s 0 , wqs ´1q 1 s › › q , sup tPr0,T s › › ›pr Č pDF s 0,t qp s 0 , wqs ´1q 1 s › › › q ă c 1 F e c 1 F p1`d r g λ px,yqq . (5.81) Proof. The C 1 regularity of s Þ Ñ pDF s 0,t q ´1 follows from that of s Þ Ñ Č pDF s 0,t q ´1 since rDF s 0,t p s 0 , wqs ´1 " pθ, ωq ´1 s 0 r Č pDF s 0,t qp s 0 , wqs ´1pθ, ωq s t and s Þ Ñ pθ, ωq ´1 s t
is C 1 . By Theorem 5.17, Č pDF s 0,t qp s 0 , wq is C 1 in s for almost all w. Hence r Č pDF s 0,t qp s 0 , wqs ´1 is also C 1 in s by the identity r Č pDF s 0,t qp s 0 , wqs ´1 ˝Č pDF s 0,t qp s 0 , wq " Id.

For (5.81), it suffices to consider the second estimation. For z 0 P T o FpR m q, let z s t´τ :" r Č pDF s 0,t´τ qp s 0 , wqs ´1z 0 , @τ P r0, ts, @s P r´s 0 , s 0 s.

It satisfies the SDE

dz s t´τ pwq " m ÿ j"1 `´M j p s τ qz s t´τ pwq d Ð Ý α s,j τ pwq `rM j p s τ qs 2 z s t´τ pwq dτ ˘`Np s τ qz s t´τ pwq dτ,
where pM j q 1ďjďm , N are given in (4.25), (4.26). For pz s t q 1 s :" pdz s t {dsq| s , its SDE is

dpz s t´τ q 1 s pwq " m ÿ j"1 `´M j p s τ qpz s t´τ q 1 s pwq d Ð Ý α s,j τ pwq `rM j p s τ qs 2 pz s t´τ q 1 s pwq dτ ˘`Np s τ qpz s t´τ q 1 s pwq dτ `m ÿ j"1
´`M j p s τ q d Ð Ý α s,j τ pwq ˘1 s z s t´τ ``m ÿ j"1 rM j p s τ qs 2 `Np s τ q ˘1 s z s t´τ dτ.

Let O s " ppO s q j l q j,lďm , g " pg s,j q jďm . They are differentiable in s by Theorem 5.17. Let `Ap1q l ˘s τ :" m ÿ j"1 ´pM j p s τ qq 1 s pO s τ q j l `Mj p s τ qppO s τ q j l q 1 s ¯, @l ď m, `Ap2q ˘s τ :"

m ÿ j"1 ´Mj p s τ qpg s,j τ q 1 s ``rM j p s τ qs 2 ˘1 s ¯``N p s τ q ˘1 s ´2 m ÿ l,j"1 M j p s τ qpO s τ q j l `Ap1q l ˘s τ .
By Duhamel's principle, we have

pz s t q 1 s " " " Č pDF s 0,t qp s 0 , wq ‰ ´1ż t 0 " Č pDF s τ,t qp s τ , wq ‰`A p1q l ˘s τ pwq " Č pDF s τ,t qp s τ , wq ‰ ´1 d Ð Ý B l τ  z 0 `"" Č pDF s 0,t qp s 0 , wq ‰ ´1 ż t 0 " Č pDF s τ,t qp s τ , wq ‰`A p2q ˘s τ pwq " Č pDF s τ,t qp s τ , wq ‰ ´1dτ  z 0 .

This means

`" Č pDF s 0,t qp s 0 , wq For (5.81), it suffices to show the same bound type is valid for pIq :" sup

‰ ´1˘1 s " ż t 0 " Č pDF s 0,τ qp s 0 , wq ‰ ´1`A p1q l ˘s τ pwq " Č pDF s τ,t qp s τ , wq ‰ ´1d Ð Ý B l τ `ż t 0 " Č pDF s 0,τ qp s 0 , wq ‰ ´1`A p2q ˘s τ pwq " Č pDF s τ,t qp s τ , wq ‰ ´1dτ ": pIq s t `pIIq s t .
sPr´s 0 ,s 0 s E P ẙ,x,T sup tPr0,T s }pIq s t } q , pIIq :" sup sPr´s 0 ,s 0 s E P ẙ,x,T sup tPr0,T s }pIIq s t } q .
This will follow from Lemma 5.27 and Proposition 4.14. Clearly,

E P ẙ,x,T sup τ Pr0,T s › › › `Apiq l ˘s τ pwq › › › q ď c M E P ẙ,x,T max sup tPr0,T s › › p s t q 1 s › › 2q , sup tPr0,T s › › pO s t q 1 s › › 2q , sup tPr0,T s › › pg s t q 1 s › › q ( ,
where c M depends on the norm bounds of tM j u and their differentials. Hence by Lemma 5.26, there are constants c A (which depends on s 0 , m, q, s and }g 0 } C 3 ) and c A (which depends on m, q, T, T 0 and }g 0 } C 3 ) such that

(5.82)

E P ẙ,x,T sup τ Pr0,T s › › › `Apiq l ˘s τ pwq › › › q ď c A e c A p1`d r g λ px,yqq . Let pIIIq s t :" ż t 0 " Č pDF s 0,τ qp s 0 , wq ı ´1`A p1q l ˘s τ pwq " Č pDF s τ,t qp s τ , wq ı ´1d Ð Ý b l τ , pIVq s t :" ż t 0 " Č pDF s 0,τ qp s 0 , wq ı ´1`A p1q l ˘s τ pwq " Č pDF s τ,t qp s τ , wq ı ´1p ´1 τ ∇ ln ppT ´τ, y 0 τ , xqq l dτ,
where b τ is the Brownian motion in Lemma 4.15 for P y,x,T . Then

E P ẙ,x,T sup tPr0,T s }pIq s t } q ď 2 q´1 E P ẙ,x,T sup tPr0,T s }pIIIq s t } q `22q´1 E P ẙ,x,T sup tPr0,T s }pIVq s t } q .
As usual, we can use Hölder's inequality and Doob's maximal inequality of sub-martingales to deduce that

`EP ẙ,x,T sup tPr0,T s }pIIIq s t } q ˘2 ďppT, x, yqp 2q 2q ´1 q 2q E P ẙ,x,T }pIIIq s T } 2q .
Let C 1 p¨q be the constant function in Lemma 4.7. We continue to compute that

E P ẙ,x,T }pIIIq s T } 2q ď C 1 p2qqE P ẙ,x,T ˇˇˇż T 0 › › › " Č pDF s 0,τ qp s 0 , wq ‰ ´1`A p1q l ˘s τ pwq " Č pDF s τ,T qp s τ , wq ‰ ´1› › › 2 dτ ˇˇˇq ď C 1 p2qqT q ˜EP ẙ,x,T sup 0ďtătďT › › rDF s t,t p s t , wqs ´1› › 8q ¨EP ẙ,x,T sup τ Pr0,T s › › › `Ap1q l ˘s τ pwq › › › 4q ¸1 2 ,
which has the same type of bound as in (5.81) by Lemma 5.27 and (5.82). Similarly,

˜EP ẙ,x,T sup tPr0,T s }pIVq s t } q ¸3ďE P ẙ,x,T sup 0ďtătďT › › rDF s t,t p s t , wqs ´1› › 6q ¨EP ẙ,x,T sup τ Pr0,T s › › › `Ap1q l ˘s τ pwq › › › 3q ¨EP ẙ,x,T ˇˇˇż T 0 }∇ ln ppT ´τ, y 0 τ , xq} dτ ˇˇˇ3 q ,
which also has the same type of bound as in (5.81) by Proposition 4.14, Lemma 5.27 and (5.82). Altogether, the same type of bound as in (5.81) is valid for pIq. This is also true for pIIq by Lemma 5.27 and (5.82) since

´EP ẙ,x,T }pIIq s T } q ¯2 ď T 2q E P ẙ,x,T sup 0ďtătďT › › ›r Č pDF s t,t qp s t , wqs ´1› › › 2q E P ẙ,x,T sup τ Pr0,T s › › › `Ap2q l ˘s τ pwq › › › 2q .
With Lemmas 5.26-5.28, we can deduce the differentiability of pDπptu T s λ q p1q 0 q ˝Fs in s. Proposition 5.29. Fix T 0 ą 0. For any q ě 1 and T ą T 0 , there are c F (depending on s 0 , m, q, s, }g 0 } C 2 and }X 0 } C 1 ) and c F (depending on s 0 , m, q, s, T, T 0 and }g 0 } C 3 ) such that

sup sPr´s 0 ,s 0 s E P ẙ,x,T › › ›pDπptu T s λ q p1q 0 q ˝Fs › › › q ď c F e c F p1`d r g px,yqq . (5.83)
The one parameter family of processes tpDπptu T s λ q p1q 0 q ˝Fs u is differentiable in s. Let ∇ s T,V,s Dπptu T s λ q p1q 0 :" ´pDπptu T s λ q p1q 0 q ˝Fs ¯1 s .

For any q ě 1 and T ą T 0 , there are c 1 F (depending on s 0 , m, q, s, }g 0 } C 3 and }X 0 } C 2 ) and c 1 F (depending on s 0 , m, q, s, T, T 0 and }g 0 } C 3 ) such that

sup sPr´s 0 ,s 0 s E P ẙ,x,T › › ›∇ s T,V,s Dπptu T s λ q p1q 0 › › › q ď c 1 F e c 1 F p1`d r g px,yqq . (5.84) Proof. Recall that pDπptu T s λ q p1q 0 q ˝Fs " Dπ `ptu T s λ q p1q 0 ˝Fs ˘" Dπ `ptu s T s λ q p1q 0 pwq ˘,
where

ptu s T s λ q p1q 0 pwq " rDF s 0,T p s 0 , wqs ´1ptu 0 s λ q p1q 0 ´ż T 0 rDF s 0,t p s 0 , wqs ´1pH λ q p1q 0 p s t , e i q ˝dÐ Ý α s,i t pwq. Let Č ptu s T s λ q p1q 0 pwq :" pθ, ωq s 0 ˆptu s T s λ q p1q 0 pwq ˙, Č ptu s 0 s λ q p1q 0 :" pθ, ωq s T ´ptu 0 s λ q p1q 0 ¯.
It is easy to obtain the following Itô form expression:

Č ptu s T s λ q p1q 0 pwq "r Č pDF s 0,T qp s 0 , wqs ´1 Č ptu s 0 s λ q p1q 0 ´ż T 0 r Č pDF s 0,t qp s 0 , wqs ´1´ `pH λ q p1q 0 p s t , e i q ˘ei dt, `pH λ q p1q 0 p s t , d Ð Ý α s t pwqq ˘¯.
For Proposition 5.29, it is equivalent to show the differentiability of s Þ Ñ Č ptu s T s λ q p1q 0 pwq and estimate the conditional L q integrals of its differential process and itself.

The estimation in (5.83) is valid since

sup sPr´s 0 ,s 0 s E P ẙ,x,T › › ›pDπptu T s λ q p1q 0 q ˝Fs › › › q ď sup sPr´s 0 ,s 0 s E P ẙ,x,T › › › › Č ptu s T s λ q p1q 0 pwq › › › › q ,
where the second term has a bound in (5.83) by following the argument of (4.45) in Proposition 4.27 and using Lemma 5.26 and Lemma 5.27.

The processes α s , s and r Č DF s 0,t p s 0 , wqs ´1 are all differentiable in s by Theorem 5.17. Lemmas 5.26-5.28 show that α s t , Υ V,α s , p s t q 1 s , r Č pDF s 0,t qp s 0 , wqs ´1 and pr Č pDF s 0,t qp s 0 , wqs ´1q 1 s all have bounded sup L q (q ě 1) norm with respect to P y,x,T . Hence s Þ Ñ Č ptu s T s λ q p1q 0 pwq is also differentiable in s and the differential is

`Č ptu s T s λ q p1q 0 pwq ˘1 s " `r Č pDF s 0,T qp s 0 , wqs ´1˘1 s Č ptu s 0 s λ q p1q 0 ´ż T 0 `r Č pDF s 0,t qp s 0 , wqs ´1˘1 s
´ `pH λ q p1q 0 p s t , e i q ˘ei dt, `pH λ q p1q

0 p s t , d Ð Ý α s t pwqq ˘ż T 0 r Č pDF s 0,t qp s 0 , wqs ´1 ´ `pH λ q p1q 0 p s t , e i q ˘1 s e i dt, `pH λ q p1q 0 p s t , ¨q˘1 s d Ð Ý α s t pwq ` `pH λ q p1q 0 p s t , dΥ V,α s q
˘":

Ipsq `IIpsq `IIIpsq.
This process has a continuous version in s by Kolmogorov's criterion (or by continuity of α s , s , Υ V,α s , p s t q 1 s and r Č pDF s 0,t qp s 0 , wqs ´1 in s using Theorem 5.17). For (5.84), we do the corresponding conditional estimations for Ipsq, IIpsq and IIIpsq. Clearly,

E P ẙ,x,T |Ipsq| q ď E P ẙ,x,T sup tPr0,T s › › ›pr Č pDF s 0,t qp s 0 , wqs ´1q 1 s › › › q ¨› › ptu 0 s λ q p1q 0 › › q ,
which, by (5.81), has a bound as in (5.84). Put

II 1 psq :" ´ż T 0 `r Č pDF s 0,t qp s 0 , wqs ´1˘1 s
´ `pH λ q p1q 0 p s t , e i q ˘ei dt, `pH λ q p1q 0 p s t , g s t pwqdtq ˘¯,

II 2 psq :" ´ż T 0 `r Č pDF s 0,t qp s 0 , wqs ´1˘1 s ´0, `pH λ q p1q 0 p s t , O s t d Ð Ý B t pwqq ˘¯.
For IIpsq, we have

E P ẙ,x,T |IIpsq| q ď 2 q´1 ´EP ẙ,x,T |II 1 psq| q `EP ẙ,x,T |II 2 psq| q ¯.
As before, we can use Hölder's inequality, Doob's inequality of submartingales and Burkholder's inequality to obtain some Cpq, T q depending on s 0 , m, q, s, T , }g 0 } C 3 and }X 0 } C 2 such that

E P ẙ,x,T |IIpsq| q ďCpq, T qT ´m 0 ˜EP ẙ,x,T sup tPr0,T s › › ›pr Č pDF s 0,t qp s 0 , wqs ´1q 1 s › › › q ÈP ẙ,x,T sup tPr0,T s › › ›pr Č pDF s 0,t qp s 0 , wqs ´1q 1 s › › › 2q E P ẙ,x,T e t2q ş T 0 }∇ ln ppT ´τ,y 0 τ ,xq} dτ u ˘1 2 ¸,
which has a bound as in (5.84) by Lemma 5.28 and Proposition 4.14. The same argument applies to IIIpsq and we obtain some C 1 pq, T q depending on s 0 , m, q, s, T , }g 0 } C 3 and }X 0 } C 2 such that

E P ẙ,x,T |IIIpsq| q ďC 1 pq, T qT ´m 0 `EP ẙ,x,T sup tPr0,T s › › ›r Č pDF s 0,t qp s 0 , wqs ´1› › › 2q ˘1 2 !1 ``E P ẙ,x,T e t2q ş T 0 }∇ ln ppT ´τ,y 0 τ ,xq} dτ u ˘1 2 ËP ẙ,x,T ż T 0 ˇˇ` `pH λ q p1q 0 p s t , e i q ˘ei ˘1 s ˇˇ2 q dt ˘1 2 ``E P ẙ,x,T ż T 0 |K V,α s | 2q dt ˘1 2 ``E P ẙ,x,T ż T 0 ˇˇ` pH λ q p1q 0 p s t , ¨q˘1 s g s t ˇˇ2 q dt ˘1 2 ˙* ,
which also has a bound as in (5.84) by Lemma 5.26, Lemma 5.27 and Proposition 4.14.

We can define `Dπptu T s λ q p1q λ ˘˝tF s s λ for all λ. Let V, F s and s be as in Section 5.2. For y P Ă M , let pty t s λ pwq, t t s λ pwqq tPr0,T s be the stochastic pair in p Ă M , O r g λ p Ă M qq which defines the r g λ -Brownian motion on Ă M starting from y. Following Theorem 5.17, we can extend the map F s on y to be a map tF s y s λ on paths pty t s λ pwqq tPr0,T s so that ty s t s λ pwq :" ´tF s y s λ ptys λ r0,T s pwqq ¯ptq, @t P r0, T s, and its horizontal lift `t s t s λ pwq ˘tPr0,T s with pt s 0 s λ q 1 s " 0 are such that d ds `ty s t s λ pwq ˘" Υ V,ty s s λ ptq " sptqt s t s λ pt s 0 s λ q ´1VpF s pyqq.

Accordingly, we denote by tα s t s λ the anti-development of ty s t s λ and let `tF s t,t s λ ˘0ătătăT be the stochastic flow map corresponding to the SDE

dβ t " H λ pβ t , ˝dtα s t s λ pwqq
with tangent maps pDtF s t,t s λ q 0ătătăT . We will omit the upper-script at s " 0. For x P Ă M , we define tF s s λ on C x pr0, T s, Ă M q conditioned on the value of β T , i.e., tF s s λ pβq :" tF s β T s λ pβq, @β P C x pr0, T s, Ă M q.

Let ptx t s λ pwq, tu t s λ pwqq tPr0,T s be the stochastic pair in p Ă M , O r g λ p Ă M qq which defines the r g λ -Brownian motion on Ă M starting from x. The correspondence rule in Corollary 5.25 shows that conditioned on tx T s λ " y, the distribution of ptu T s λ q p1q λ pwq is the same as, conditioned on ty T s λ " x, the distribution of ptu T s λ q p1q λ pwq :"rDtF 0,T s λ pt 0 s λ , wqs ´1ptu 0 s λ q

p1q λ ´ż T 0 rDtF 0,t s λ pt 0 s λ , wqs ´1pH λ q p1q λ `t t s λ , ˝dB t pwq ˘,
where ˝dÝ Ñ B t pwq is the backward Stratonovich infinitesimal. Then we define

pDπptu T s λ q p1q λ q ˝tF s s λ :" Dπ `ptu T s λ q p1q λ ˝tF s s λ ˘" Dπ `ptu s T s λ q p1q λ pwq ˘,
where ptu s T s λ q p1q λ pwq " rDtF s 0,T s λ pt s 0 s λ , wqs ´1ptu 0 s λ q p1q λ ´ż T 0 rDtF s 0,t s λ pt s 0 s λ , wqs ´1pH λ q p1q λ `t s t s λ , ˝dtα s t s λ pwq ˘. (5.85)

The proof of Proposition 5.29 works for tF s s λ , which gives the following.

Proposition 5.30. For each λ, the one parameter family of processes tpDπptu T s λ q p1q λ q tF s s λ u is differentiable in s. Moreover, pDπptu T s λ q p1q λ q˝tF s s λ and the differential stochastic process

∇ s,λ T,V,s Dπptu T s λ q p1q λ :" ´pDπptu T s λ q p1q λ q ˝tF s s λ ¯1 s
conditioned on tx T s λ " y are L q (q ě 1) integrable, locally uniformly in the s parameter.

For later use, we list and reformulate some differentials related to ∇ T,V,s Dπptu T s λ q p1q λ . The upper-scripts λ in ∇ λ , R λ , Ric λ , θ λ , λ and pθ, q λ are to indicate the metric r g λ used.

Lemma 5.31. We have the following for almost all w and for all t P r0, T s. i)

ptα s t s λ q 1 0 " Υ λ V,B ptq
:" ż t 0 ´pt 0 s λ q ´1ps 1 pτ qVpyqq ´Ric λ `ΥV,tys λ pτ q ˘¯dτ ´ż t

0 xK λ V,B pτ q, dB τ y,
where Υ V,tys λ pτ q :" spτ qt τ s λ pt 0 s λ q ´1Vpyq and

K λ V,B pτ q " ż τ 0 pt r τ s λ q ´1R λ ´t r τ s λ dB r τ , Υ V,tys λ pr τ q ¯t r τ s λ `ż τ 0 pt r τ sq ´1p∇ λ pt r τ s λ e i qR λ q ´t r τ s λ e i , Υ V,tys λ pr τ q ¯t r τ s λ dr τ . ii) pθ, q λ t ts λ pt s t s λ q 1 0 " `sptqpt 0 s λ q ´1Vpyq, K λ V,B ptq ˘. iii) For s Þ Ñ v s t P HT t s t s λ Fp Ă M q, let v s τ :" rDtF s τ,t s λ pt s τ s λ , wqs ´1v s t , τ P r0, ts. Then `vs 0 ˘1 0 " ż t 0 " DtF 0,τ s λ `t 0 s λ , w ˘‰´1 ´fλ `vτ , pt s τ s λ q 1 0 ˘¯, where f λ `vτ , pt s τ s λ q 1 0 ˘"∇ p2q,λ `vτ , pt s τ s λ q 1 0 ˘Hλ pt τ s λ , ˝dB τ q `∇λ pv τ qH λ `t τ s λ , ˝dΥ λ V,B pτ q Ȓλ `Hλ pt τ s λ , ˝dB τ q, pt s τ s λ q 1 0 ˘vτ . (5.86) The Itô form of `vs 0 ˘1 0 in pθ, q-chart is pθ, q λ t 0 s λ `vs 0 ˘1 0 " ż t 0 " Č DtF 0,τ s λ `t 0 s λ , w ˘ı´1 ! pθ, q λ f λ I pv τ , pt s τ s λ q 1 0 q ``r f θ,λ A pv τ , pt s τ s λ q 1 0 q, r f ,λ A pv τ , pt s τ s λ q 1 0 q ˘) ,
where f λ I pv τ , pt s τ s λ q 1 0 q is f λ pv τ , pt s τ s λ q 1 0 q with the Stratonovich infinitesimals ˝dB τ , ˝dΥ λ V,B pτ q replaced by the Itô infinitesimals dB τ , dΥ λ V,B pτ q,

r f θ,λ A `vτ , p s τ q 1 0 ˘" λ ´f`v τ , pt s τ s λ q 1 0 , e i ˘¯e i dτ `θλ ´"Hp τ , e i q, f λ `vτ , pt s τ s λ q 1 0 , e i ˘ı¯d τ, r f ,λ A `vτ , pt s τ s λ q 1 0 ˘" pt τ s λ q ´1R λ ´t τ s λ e i , t τ s λ θ λ pf λ pv τ , pt s τ s λ q 1 0 , e i qq ¯t τ s λ dτ ` λ
´"H λ pt τ s λ , e i q, f λ `vτ , pt s τ s λ q 1 0 , e i ˘ı¯d τ, and

f λ `vτ , pt s τ s λ q 1 0 , e i ˘" ∇ p2q,λ `vτ , pt s τ s λ q 1 0 ˘Hλ pt τ s λ , e i q `Rλ `Hλ pt τ s λ , e i q, pt s τ s λ q 1 0 ˘vτ `∇λ pv τ qH λ `t τ s λ , K λ V,B e i ˘.
Proof. Without loss of generality, we can consider the case λ " 0. The i), ii) are straight forward consequences of Theorem 5.17 reporting α 0 " B in the formulas in Lemma 5.12 and Corollary 5.10. For iii), a comparison of the SDEs (5.41), (5.43) in Lemma 5.14 with that of the tangent maps rDF 0,τ s ´1, r Č DF 0,τ s ´1 shows that we can use Duhamel's principle to formulate pv s 0 q 1 0 , pθ, q 0 pv s 0 q 1 0 as above. Proposition 5.32. With all the notations as above, then

ˆptu s T s λ q p1q λ pwq ˙1 0 " ż T 0 rDtF 0,τ s λ pt 0 s λ , wqs ´1 ! f λ `pt τ s λ q p1q λ , pt s τ s λ q 1 0 ∇λ ppt s τ s λ q 1 0 qpH λ q p1q λ pt τ s λ , ¨q˘˝d B τ ´pH λ q p1q λ `t τ s λ , ˝dΥ λ V,B pτ q ˘) , (5.87)
where f λ `pt τ s λ q p1q λ , p s τ q 1 0 ˘is as in (5.86) replacing v τ by pt τ s λ q p1q λ and pt τ s λ q p1q λ pwq :"

" DtF τ,T s λ pt 0 s λ , wq ‰ ´1ptu 0 s λ q p1q λ ´ż T τ " DtF τ,t s λ pt τ s λ , wq ‰ ´1pH λ q p1q λ `t t s λ , ˝dB t pwq ˘.
In pθ, q λ -chart, we have the Itô integral expression

ˆpθ, ωq λ s 0 `ptu s T s λ q p1q 0 pwq ˘˙1 0 " ż T 0 " Č DtF 0,τ s λ `t 0 s λ , w ˘ı´1 ! pθ, q λ f λ I `pt τ s λ q p1q λ , pt s τ s λ q 1 0 ȓ f θ,λ A ppt τ s λ q p1q λ , pt s τ s λ q 1 0 q, r f ,λ A ppt τ s λ q p1q λ , pt s τ s λ q 1 0 q ¯ λ `∇λ ppt s τ s λ q 1 0 qpH λ q p1q λ pt τ s λ , e i q ˘ei dτ, λ `∇λ ppt s τ s λ q 1 0 qpH λ q p1q λ pt τ s λ , dB τ q ˘` λ `pH λ q p1q λ pt τ s λ , dΥ λ V,B pτ qq ˘¯) .
Proof. Differentiating (5.85), we obtain

ˆptu s T s λ q p1q λ pwq ˙1 0 " `rDtF s 0,T s λ pt s 0 s λ , wqs ´1˘1 0 ptu 0 s λ q p1q λ ´ż T 0 `rDtF s 0,t s λ pt s 0 s λ , wqs ´1˘1 0 pH λ q p1q λ `t t s λ , ˝dB t pwq ż T 0 rDtF 0,t s λ pt 0 s λ , wqs ´1`∇ λ ppt s t s λ q 1 0 qpH λ q p1q λ pt t s λ , ¨q˘˝d B t pwq ´ż T 0 rDtF 0,t s λ pt 0 s λ , wqs ´1pH λ q p1q λ `t t s λ , ˝dΥ λ V,B pt, wq ":
pIq `pIIq `pIIIq `pIVq.

By iii) of Lemma 5.31, we have pIq "

ż T 0 " DtF 0,τ s λ pt 0 s λ , wq ‰ ´1 f λ ´"DtF τ,T s λ pt 0 s λ , wq ‰ ´1ptu 0 s λ q p1q λ , pt s τ s λ q 1 0 ¯.
For 0 ď τ ă t ď T , put v τ,t :" " DtF τ,t s λ pt τ s λ , wq ‰ ´1pH λ q p1q λ `t t s λ , ˝dB t pwq ˘.

We continue to compute that

pIIq " ´ż T 0 ż t 0 " DtF 0,τ s λ pt 0 s λ , wq ‰ ´1 f λ `vτ,t , pt s τ s λ q 1 0 " ´ż T 0 " DtF 0,τ s λ pt 0 s λ , wq ‰ ´1 f λ `ż T τ v τ,t , pt s τ s λ q 1 0 ˘.
Altogether, we obtain pIq `pIIq "

ż T 0 rDtF 0,τ s λ pt 0 s λ , wqs ´1 f λ `pt τ s λ q p1q λ , pt s τ s λ q 1 0 ˘.
Hence (5.87) holds true. The Itô form integral expression of `pθ, ωq s 0 `ptu s T s λ q p1q 0 pwq ˘˘1 0 can be obtained using the Itô form in iii) of Lemma 5.31.

As a corollary of Proposition 5.32, we can further express the differential pptu T s λ q p1q λ ˝tF s s λ q 1 0 ": ´ptu s T s λ q p1q λ pwq ¯1 0 using ptu t s λ pwqq tPr0,T s and the tangent maps trDt Ý Ñ F t,t s λ ptu t s λ , wqsu 0ďtătďT of the flow maps tt Ý Ñ F t,t s λ ptu t s λ , wqu 0ďtătďT associated to (5.73). We only give the Stratonovich form. Let K λ,u V,B pt, wq :" ż t 0 ptu T ´r τ s λ q ´1R λ ´ptu T ´r τ s λ q ´1dB T ´r τ pwq, tu T ´r τ s λ ptu T s λ q ´1rspr τ qVpyqs ¯tu T ´r τ s λ `ż t 0 ptu T ´r τ s λ q ´1p∇ λ ptu T ´r τ s λ e i qR λ q ´tu T ´r τ s λ e i , tu T ´r τ s λ ptu T s λ q ´1rspr τ qVpyqs ¯tu T ´r τ s λ dr τ , Υ λ,u V,B pτ q :" ż T ´τ 0 `ptu T s λ q ´1ps 1 ptqVpyqq ´Ric λ `tu T ´ts λ ptu 0 s λ q ´1rsptqVpyqs ˘˘dt ´ż T ´τ 0 xK λ,u V,B pt, wq, dB t y.

Then Υ λ,u V,B pτ q corresponds to Υ λ V,B pT ´τ q and they have the same distribution. Put ptu s τ,T s λ q 1 0 :" pθ, q ´1 tuτ s λ ´spT ´τ qptu T s λ q ´1Vpyq, K λ,u V,B pT ´τ q ¯.

Corollary 5.33. With all the notations as above,

´ptu s T s λ q p1q λ pwq ¯1 0 " ż T 0 " Dt Ý Ñ F τ,T s λ ptu τ s λ , wq ‰ ! f λ,u `ptu τ s λ q p1q λ , ptu s τ,T s λ q 1 0 ∇λ pptu s τ,T s λ q 1 0 qpH λ q p1q λ ptu τ s λ , ¨q˘˝d B τ pwq `pH λ q p1q λ `tu τ s λ , ˝dΥ λ,u V,B pτ q ˘) ,
where f λ,u `ptu τ s λ q p1q λ , ptu s τ,T s λ q 1 0 ˘" ´∇λ,p2q `ptu τ s λ q p1q λ , ptu s τ,T s λ q 1 0 ˘Hλ `tu τ s λ , ˝dB τ pwq ∇λ pptu τ s λ q p1q λ qH λ `tu τ s λ , ˝dΥ λ,u V,B pτ q Ȓλ `Hλ ptu τ s λ , ˝dB τ pwqq, ptu s τ,T s λ q 1 0 ˘ptu τ s λ q p1q λ .

In pθ, q λ -chart, we have the Itô integral expression

´pθ, qptu s T s λ q p1q λ pwq ¯1 0 " ż T 0 " D Č t Ý Ñ F τ,T s λ ptu τ s λ , wq ‰ ! pθ, q λ f λ,u I `ptu τ s λ q p1q λ , ptu s τ,T s λ q 1 0 ȓ f θ,λ A `ptu τ s λ q p1q λ , ptu s τ,T s λ q 1 0 ˘, r f ,λ
A `ptu τ s λ q p1q λ , ptu s τ,T s λ q 1 0 ˘¯ λ `∇λ pptu s τ,T s λ q 1 0 qpH λ q p1q λ ptu τ s λ , e i q ˘ei dτ, λ `∇λ pptu s τ,T s λ q 1 0 qpH λ q p1q λ ptu τ s λ , dB τ q ˘` λ `pH λ q p1q λ `tu τ s λ , dΥ λ,u V,B pτ q ˘˘¯) .

Proof. Note that `ptu s T s λ q p1q λ pwq ˘1 0 conditioned on tx T s λ " y is the same as `ptu s T s λ q p1q λ pwq ˘1 0 conditioned on ty T s λ " x. The formulas follow by Proposition 5.32 using the correspondence between " Dt Ý Ñ F t,t s λ ptu t s λ , wq ‰ and " DtF T ´t,T ´ts λ pt T ´ts λ , wq ‰ ´1.

5.6. The differential of λ Þ Ñ p λ pT, x, ¨q. We will show Theorem 5.1 in two steps, namely, the k " 3 and k ą 3 cases. We begin with the k " 3 case. As we sketched in Section 5.1, the strategy is to show z λ,1 T defined in (5.15) is a C 1 vector field, then derive a conditional path-wise formula of Div λ z λ,1

T pyq and use it to give the estimation in (5.2).

Lemma 5.34. Let λ P p´1, 1q Þ Ñ g λ P M 3 pM q be a C 3 curve. Let

x P Ă M , T P R `. The map Φ 1 λ : Y Þ Ñ Φ 1 λ pY q defined in (5.13) is a locally bounded C 1 functional on C k bounded vector fields Y on Ă M . Consequently, tz λ,1 T pyqu is a C 1 vector field on Ă M . Proof. Recall that Φ 1 λ pY qpyq " E ´@Y ptx T s λ pwqq, Dπpu T s λ q p1q λ pwq D λ ˇˇtx T s λ pwq " y ¯" @ Y pyq, z λ,1 T pyq D .
Hence,

› › Φ 1 λ p¨qpyq › › ď › › z λ,1 T pyq › › ď E P λ x,y,T › › ptu T s λ q p1q λ pwq › › " 1 ppT, x, yq E P λ,x ,y,T › › ptu T s λ q p1q λ pwq › › .
By Proposition 4.27, there are c λ Φ (depending on }g λ } C 2 and }X λ } C 1 ) and c λ Φ (depending on T, T 0 and }g λ } C 3 ) such that

› › Φ 1 λ p¨qpyq › › ď 1 ppT, x, yq E P λ,ẙ ,x,T › › ptu T s λ q p1q λ pwq › › ď 1 ppT, x, yq c λ Φ e c λ Φ p1`d r g λ px,yqq ,
where the last term is locally uniformly bounded in the y-coordinate. This shows the map where Φ 1 λ pY, wq " @ Y ptx T s λ pwqq, Dπptu T s λ q p1q λ pwq D λ . By Proposition 5.30, the process Φ 1 λ pY, wq ˝tF s s λ is differentiable in s with (5.88)

Y Þ Ñ Φ 1 λ pY q is locally bounded. To show Φ 1 λ is C 1 , it
pΦ 1 λ ˝tF s s λ q 1 s " @ ∇ Vptx s T s λ q Y ptx s T s λ q, Dπptu s T s λ q p1q λ D λ `@Y ptx s T s λ q, ∇ s,λ T,V,s Dπptu T s λ q p1q λ D λ
and this differential is L q integrable conditioned on x T " y for every q ě 1, locally uniformly in the s parameter. By Lemma 5. T are L q integrable for all q ě 1, locally uniformly in the s parameter. Using Hölder's inequality, we conclude that

´Φ1 λ ˝tF s s λ ¨`dP λ x ˝tF s s λ {dP λ x

˘¯1

s is also L q integrable for every q ě 1, locally uniformly in the s parameter. This allows us to take the differential in s under the expectation sign of the expression of Φ 1 λ pY qpF s yq. In particular, this shows s Þ Ñ Φ 1 λ pY qpF s yq is differentiable at s " 0. Let us derive a formula for pΦ 1 λ pY qpF s yqq 1 0 . Note that tg 0 s λ " 0 and ptg s t s λ q 1 0 pwq " pt 0 s λ q ´1 " s 1 ptqVpyq ‰ ´Ric λ ´t t s λ pt 0 s λ q ´1rsptqVpyqs

¯.

Using the correspondence between t t s λ pwq conditioned on y λ T " x and tu T ´ts λ pwq conditioned on x λ T " y, we have the distribution of E 0 T under P λ y,x,T is the same as ´@∇ Vptx T s λ q Y, Dπptu T s λ q

Ð Ý E T,V,s " ´1 2 ż T 0 @ s 1 pT ´tqptu T s λ q ´1Vptx T s λ q ´Ricptu t s λ ptu T s λ q ´1spT ´tqVptx T s λ qq, d Ð Ý B t D .
p1q λ D λ `@Y ptx T s λ q, ∇ λ T,V,s Dπptu T s λ q p1q λ D λ `@Y ptx T s λ q, Dπptu T s λ q p1q λ D λ Ð Ý E T,V,s
": (5.89) where we omit the upper-script 0 of x, u and ∇ T,V,s at s " 0 for simplicity.

E P λ x,y,T `Ψ1 λ pY, Vqpwq ˘,
To show pΦ 1 λ pY qpF s yqq 1 0 is continuous in y, we compare it with its value at nearby points. Choose another smooth bounded vector field W on Ă M and let r F be the flow it generates, where we use the left upper script to indicate the parameter associated with W. As before, we can extend r F to be a one parameter family of maps t r Fs λ " tt r F y s λ u on r g λ -Brownian paths starting from x up to time T . Let t r αs λ , t r Os λ , t r gs λ , t r gs λ , t r ys λ , t r s λ and pt r s λ q 1 r denote the corresponding stochastic processes of t r Fs λ in Theorem 5.17. Then a change of variable argument for (5.89) with t r Fs λ shows that for z " r F pyq, ˝tr Fs λ ´∇λ T,V,s Dπptu T s λ q p1q λ ˇˇ2 ˙Ñ 0, r Ñ 0.

Let

E 0 T ˝tr Fs λ " ´1 2 ż T 0 @ pt r 0 s λ q ´1" s 1 ptqVpyq ‰ ´Ric λ `tr t s λ pt r 0 s λ q ´1rsptqVpyqs ˘, d ÐÝÝÝ t r α t s λ D ": ´1 2 ż T 0 @ r ptg s t s λ q 1 0 pwq, d ÐÝÝÝ t r α t s λ q D .
For r pIIIq, we have

2 ¨rpIIIq " 2 ¨EP λ y,x,T ´ˇE 0 T ˝tr Fs λ ´E0 T ˇˇ4 ď E P λ y,x,T ˇˇş T 0 @ r ptg s t s λ q 1 0 pwq, `Ð ÝÝÝ Ý t r O t s λ ´Id ˘dB t pwq `Ð ÝÝ Ý t r g t s λ dt D ˇˇ4
`EP λ y,x,T ˇˇş T 0 @ r ptg s t s λ q 1 0 pwq ´0ptg s t s λ q 1 0 pwq, dB t pwq D ˇˇ4

": r pIIIq 1 `rpIIIq 2 .

For r pIIIq 1 , the usual argument using Lemma 4.15 and Burkholder's inequality shows

3 ´3 ¨rpIIIq 1 ď E P λ y,x,T ˇˇˇż T 0 › › Ð ÝÝÝ Ý t r O t s λ ´Id › › 2 › › r ptg s t s λ q 1 0 pwq › › 2 dt ˇˇˇ2 `EP λ y,x,T ˇˇˇż T 0 2 › › r ptg s t s λ q 1 0 pwq › › › › Ð ÝÝÝ Ý t r O t s λ ´Id › › › › ∇ λ ln p λ pT ´t, ty t s λ pwq, xq › › dt ˇˇˇ4 `EP λ y,x,T ˇˇˇż T 0 › › r ptg s t s λ q 1 0 pwq › › › › Ð ÝÝ Ý t r g t s λ › › dt ˇˇˇ4 .
Note that there is some constant C which depends on }g λ } C , s and supt}V}u such that ˇˇÐ ÝÝ Ý t r g t s λ pwq ˇˇ, ˇˇr ptg s t s λ q 1 0 pwq ˇˇď Cr. Hence

3 ´3 ¨rpIIIq 1 ď pCrq 8 T 4 `pCrq 4 T 2 E P λ y,x,T sup tPr0,T s › › Ð ÝÝÝ Ý t r O t s λ ´Id › › 4 `p2Crq 4 ˜EP λ y,x,T sup tPr0,T s › › Ð ÝÝÝ Ý t r O t s λ ´Id › › 8 ¨EP λ y,x,T e 8 ş T 0 › › ∇ λ ln p λ pT ´t,tyts λ ,xq › › dt ¸1 2 .
By Lemma 5.26 and Lemma 5.28, for any q ě 1, there is some C 1 pq, T q such that

E P λ y,x,T sup tPr0,T s › › ›t r α t s λ ´tα t s λ › › › q ď C 1 pq, T qr q .
Using this and (4.34), we conclude that r pIIIq 1 Ñ 0 as r Ñ 0. Similarly, using (4.37), Burkholder's inequality and (4.34), we obtain some C 2 depending on T, dpx, yq such that

r pIIIq 2 ď E P λ y,x,T ˆż T 0 › › r ptg s t s λ q 1 0 ´ptg s t s λ q 1 0 › › 2 dt ˙2 `EP λ y,x,T ˇˇˇż T 0
A´r ptg s t s λ q 1 0 ´ptg s t s λ q 1 0 ¯, 2pt t s λ q ´1∇ λ ln p λ pT ´t, ty t s λ , xq

E λ dt ˇˇˇ4 ď C 2 E P λ y,x,T ˜sup tPr0,T s › › r ptg s t s λ q 1 0 ´ptg s t s λ q 1 0 › › 4 `sup tPr0,T s › › r ptg s t s λ q 1 0 ´ptg s t s λ q 1 0 › › 8 ¸.
The argument in Lemma 5.26 shows there is some C 3 depending on }g λ } C 3 such that

E P λ y,x,T sup tPr0,T s › › r ptg s t s λ q 1 0 ´ptg s t s λ q 1 0 › › q ď C 3 ¨EP λ y,x,T sup tPr0,T s › › t r t s λ ´t t s λ › › q ď C 3 r q sup ıPr´r 0 ,r 0 s E P λ y,x,T sup tPr0,T s › › pt r t s λ q 1 ı › › q .
This immediately implies that lim rÑ0 r pIIIq 2 " 0. For r pIVq, we have

r pIVq ď Const. ¨EP λ y,x,T ˜ˇˇˇp t r u T s λ q p1q 0 ´ptu T s λ q p1q 0 ˇˇˇ4 ¸.
Note that pθ, ωq λ t r 0 s λ pt r u T s λ q p1q λ " rD Č t r F 0,T s λ pt r 0 s λ , wqs ´1 Č ptu 0 s λ q p1q λ `ż T 0 rD Č t r F 0,t s λ pt r 0 s λ , wqs

´1

´ `pH λ q p1q λ pt r t s λ , e i q ˘ei dτ, `pH λ q p1q λ pt r t s λ , d

Ð ÝÝ Ý t r αs λ t q ˘¯.
By Lemma 5.26 and Lemma 5.27, for any q ě 1, sup rPr´r 0 ,r 0 s

E P λ y,x,T sup tPr0,T s › › ›t r αs λ t › › › q , › › ›r Č Dt r F 0,t s λ pt r 0 s λ , wqs ´1› › › q ă r ce r cp1`d r g λ px,yqq ,
where r c depends on r 0 , m, q, s and }g λ } C 2 , and r c depends on r 0 , m, q, s, T, T 0 and }g λ } C 3 . Moreover, by Lemma 5.26 and Lemma 5.28,

E P λ y,x,T sup tPr0,T s › › ›t r α t s λ ´tα t s λ › › › q ď C 4 r q , E P λ y,x,T sup tPr0,T s › › ›r Č Dt r F 0,t s λ pt r 0 s λ , wqs ´1 ´r Č DtF 0,t s λ pt 0 s λ , wqs ´1› › › q ď C 5 r q ,
where the constants C 4 , C 5 depend on }g λ } C 2 . Again, a standard split argument using these estimations and Hölder's inequality gives lim rÑ0 r pIVq " 0. To conclude that lim rÑ0 r pVq " 0, we see from (5.87) that it suffices to show for any q ą 1,

E P λ y,x,T sup tPr0,T s › › ›A t ˝tr Fs λ ´At › › › q ď C A r q
for some C A depending on }g λ } C 3 , }X } C 2 , T and dpx, yq, where A t " ptg s t s λ q 1 0 , ptO s t s λ q 1 0 , pt s t s λ q 1 0 , pt t s λ q p1q λ or r Č pDF 0,t qpt 0 s λ , wqs ´1. Using Lemma 5.11, this can be be reduced to the cases that A t " tαs λ t , t t s λ or r Č DtF 0,t s λ pt 0 s λ , wqs ´1, which were shown as above.

Let C 1 be a bound of |dVol λ ˝rF {dVol λ pyq| for r P r´r 0 , r 0 s. By using (5.69), we obtain xpt r g t s λ q 1 r pwq, dB t pwqy `1 2 ż T 0 xpt r g t s λ q 1 r pwq, t r g t s λ pwqy dt ":

r E T pwq ¨rE T pwq.

The usual argument using Lemma 4.15 and Burkholder's inequality shows that for every q ě 1, E P λ y,x,T ˇˇr E T pwq ˇˇq is locally uniformly bounded in r. Hence r pVIq Ñ 0 as r Ñ 0.

Altogether, we have shown the map Y Þ Ñ Φ 1 λ pY q is a C 1 locally bounded functional on C k vector fields Y on Ă M . Hence there exists some C 1 vector field

Ą z λ,1 T on Ă M such that Φ 1 λ pY qpyq " @ Y pyq, Ą z λ,1 T pyq D λ .
This shows Ą z λ,1 t pyq " z λ,1 T pyq. Thus tz λ,1 T pyqu forms a C 1 vector field on Ă M as claimed.

Lemma 5.35. Let λ P p´1, 1q Þ Ñ g λ P M 3 pM q be a C 3 curve. Let x P Ă M , T P R `. For any smooth bounded vector field V on Ă M , let s, ∇ λ T,V,s ,

Ð Ý E T,V,s be as above, then (5.91)

∇ λ Vpyq z λ,1 T pyq " E ´∇λ T,V,s Dπptu T s λ q p1q λ pwq `Dπptu T s λ q p1q λ pwq Ð Ý E T,V,s pwq ˇˇtx T s λ pwq " y ¯.
As a consequence,

Div λ z λ,1 T pyq " E ˆtr ´V Þ Ñ ∇ λ T,V,s Dπptu T s λ q p1q λ ¯´@ Dπptu T s λ q p1q λ , 1 2 tu T s λ ż T 0 s 1 pT ´τ qd Ð Ý B τ D λ `@Dπptu T s λ q p1q λ , 1 2 ż T 0 spT ´τ qtu T s λ ptu τ s λ q ´1pRic λ tuτ s λ q ´1d Ð Ý B τ D λ ˇˇˇt x T s λ pwq " y ˙.
Proof. Let Y be a C k bounded vector field on Ă M . By Lemma 5.34,

(5.92) Φ

1 λ pY qpyq " @ Y pyq, z λ,1 T pyq D λ ,
where all the variables Φ 1 λ pY q, Y and z λ,1 T are C 1 in y. Hence

∇ λ V pΦ 1 λ pY qqpyq " @ ∇ λ V Y pyq, z λ,1 T pyq D λ `@Y pyq, ∇ λ V z λ,1 T pyq D λ . (5.93)
Let tF s u sPR be the flow generated by a smooth vector field V. Then

∇ λ V pΦ 1 λ pY qqpyq " ´Φ1 λ pY qpF s yq ¯1 0 .
It was shown in Lemma 5.34 that

´Φ1 λ pY qpF s yq ¯1 0 " E ´@∇ λ Vptx T s λ q Y, Dπptu T s λ q p1q λ D λ `@Y ptx T s λ q, ∇ λ T,V,s Dπptu T s λ q p1q λ D λ `@Y ptx T s λ q, Dπptu T s λ q p1q λ D λ Ð Ý E T,V,s ˇˇtx T s λ pwq " y ¯. (5.94) Applying (5.92) for the C k´1 vector field ∇ λ Y (instead of Y ) gives E ´@∇ λ Vptx T s λ q Y, Dπptu T s λ q p1q λ D λ ˇˇtx T s λ pwq " y ¯" @ ∇ λ V Y pyq, z λ,1 T pyq D λ .
Report this in (5.94) and then compare it with (5.93). We obtain

@ Y pyq, ∇ λ V z λ,1 t pyq D λ " E ´@Y ptx T s λ q, ∇ λ T,V,s Dπptu T s λ q p1q λ D λ `@Y ptx T s λ q, Dπptu T s λ q p1q λ D λ Ð Ý E T,V,s ˇˇtx T s λ pwq " y " A Y pyq, E ´∇λ T,V,s Dπptu T s λ q p1q λ pwq `Dπptu T s λ q p1q λ pwq
Ð Ý E T,V,s pwq ˇˇtx T s λ pwq " y ¯Eλ .

This implies (5.91) since Y is arbitrary.

The divergence pDiv λ z λ,1 T pyqq is just the trace of the mapping Vpyq

Þ Ñ ∇ λ Vpyq z λ,1 T pyq. Put Ð Ý E 1 T,V,s " ´1 2 ż T 0 @ s 1 pT ´τ qptu T s λ q ´1Vptx T s λ q, d Ð Ý B τ D , Ð Ý E 2 T,V,s " 1 2 ż T 0 @ Ric λ tuτ s λ ptu τ s λ ptu T s λ q ´1spT ´τ qVptx T s λ qq, d Ð Ý B τ D .
Then

pDiv λ z λ,1 T pyqq " E ´tr ´V Þ Ñ ∇ λ T,V,s Dπptu T s λ q p1q λ pwq ¯2 ÿ i"1 tr ´V Þ Ñ Dπptu T s λ q p1q λ pwq Ð Ý E i T,V,s pwq ¯ˇˇˇˇt x T s λ pwq " y ¸.
Take V 1 , ¨¨¨, V m to be orthogonal at y in the metric r g λ . We obtain

tr ´V Þ Ñ Dπptu T s λ q p1q λ pwq Ð Ý E 1 T,V,s pwq ¯" m ÿ j"1 @ Dπptu T s λ q p1q λ pwq, V i D λ ¨Ð Ý E 1 T,V i ,s pwq " @ Dπptu T s λ q p1q λ pwq, ´1 2 u T ż T 0 s 1 pT ´τ qd Ð Ý B τ D λ ,
Note that tu τ s λ ptu T s λ q ´1 is the backward parallel transportation along txs λ rτ,T s pwq which preserves the inner-product. Using (4.12), we obtain

Ð Ý E 2 T,V,s " 1 2 ż T 0 @ Vptx T s λ q, spT ´τ qtu T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d Ð Ý B τ D λ and tr ´V Þ Ñ Dπptu T s λ q p1q λ pwq Ð Ý E 2 T,V,s pwq " ř m j"1 xDπptu T s λ q p1q λ pwq, V i y ¨Ð Ý E 2 T,V i ,s pwq " @ Dπptu T s λ q p1q λ pwq, 1 2 ş T 0 spT ´τ qtu T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d Ð Ý B τ D λ .
Proof of Theorem 5.1 (k " 3). Let x P Ă M and T P R `. Let ptx t s λ , tu t s λ q tPR `be the stochastic process pair which defines the Brownian motion on p Ă M , r g λ q starting from x. By Lemma 4.17 and Proposition 4.27, it is true that for any f

P C 8 c p Ă M q, ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙p1q λ " ż Ă M @ ∇ λ y f pyq, z λ,1
T pyq ¨pλ pT, x, yq D λ dVol λ pyq.

Since tz λ,1 T pyqu is a C 1 vector field on Ă M by Lemma 5.34, the classical integration by parts argument in Section 5.1 shows that

ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙p1q λ " ´ż Ă M f pyq ´pDiv λ z λ,1
T pyqq ¨pλ pT, x, yq `@z λ,1 T pyq, ∇ λ p λ pT, x, yq

D λ ¯dVol λ pyq " ż Ă M
f pyqφ 1 λ pT, x, yqp λ pT, x, yq dVol λ pyq.

The function φ 1 λ pT, x, yq is continuous in y, uniformly in λ (see Lemma 5.34). Hence its continuity in λ follows from the continuity in λ of `ş Ă M f pyqp λ pT, x, yq dVol λ pyq ˘p1q λ for any f P C 8 c p Ă M q, which is true by (5.10) and the convergence in λ of tx T s λ pwq and tu T s λ pwq in the L q -norm for every q ě 1. So the first part argument in the proof of Lemma 5.5 works, which shows that λ Þ Ñ p λ pT, x, ¨q is C 1 , the differential pp λ q p1q λ pT, x, yq is continuous in y and pp λ q p1q λ pT, x, yq ¨ρλ pyq `pλ pT, x, yq ¨pρ λ q p1q λ pyq " φ 1 λ pT, x, yqp λ pT, x, yqρ λ pyq. This gives (5.1) since ρ λ is non-zero for V g small. Next, we show (5.2) with l " 0. For this, it suffices to show the same type of bound holds for the L q -norm of φ 1 λ pT, x, yq. Note that, by Lemma 5.34, z λ,1 T pyq is such that @ z λ,1 T pyq, ∇ λ ln p λ pT, x, yq

D λ " E ´@Dπpu T s λ q p1q λ pwq, ∇ λ ln p λ pT, x, tx T s λ pwqq D λ ˇˇtx T s λ " y ¯.
Using this and the formula of Div λ z λ,1 T in Lemma 5.35, we obtain (5.95) φ 1 λ pT, x, yq " E ´r φ 1 λ pT, x, wq ˇˇtx T s λ pwq " y ¯,

where

r φ 1 λ pT, x, wq " ´tr `V Þ Ñ ∇ λ T,V,s Dπptu T s λ q p1q λ ˘`xDπptu T s λ q p1q λ , 1 2 tu T s λ ż T 0 s 1 pT ´τ qd Ð Ý B τ D λ ´@Dπptu T s λ q p1q λ , 1 2 ż T 0 spT ´τ qtu T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d Ð Ý B τ D λ (5.96)
´@Dπptu T s λ q p1q λ , ∇ λ ln p λ pT, x, tx T s λ q D λ ": pIqpT, x, wq `pIIqpT, x, wq `pIIIqpT, x, wq `pIVqpT, x, wq.

So,

› › φ 1 λ pT, x, ¨q› › q L q " ż Ă M
ˇˇE `r φ 1 λ pT, x, wq ˇˇtx T s λ pwq " y ˘ˇˇq p λ pT, x, yq dVol λ pyq ď ż Ă M E `} r φ 1 λ pT, x, wq} q ˇˇtx T s λ pwq " y ˘pλ pT, x, yq dVol λ pyq ď 4 q´1 `E}pIq} q `E}pIIq} q `E}pIIIq} q `E}pIVq} q ˘.

Hence we will obtain (5.2) with l " 0 if pIq, pIIq, pIIIq and pIVq all have the same type of L q bounds. This actually follows from Proposition 4.11 and Proposition 4.27. For pIVq, it is true by (4.44) and (4.33) since

`E }pIVq} q ˘2 ď E › › ›ptu T s λ q p1q λ › › › 2q ¨E › › ›∇ λ ln p λ pT, x, tx T s λ q › › › 2q .
Using Hölder's inequality, we obtain

`E }pIIIq} q ˘2 ď E › › ›ptu T s λ q p1q λ › › › 2q ¨E › › › › 1 2 ż T 0 spT ´τ qtu T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d Ð Ý B τ › › › › 2q .
Using Proposition 4.27 and Lemma 4.7, it is easy to show that E }pIIIq} q has the same bound type in (5.2) with l " 0. The term pIIq can be handled in the same way. For pIq, it suffices to estimate the L q -norm of `ptu s T s λ q p1q λ pwq ˘1 0 for V with norm 1. Split the Itô integral of `pθ, qptu s T s λ q p1q λ pwq ˘1 0 in Corollary 5.33 with infinitesimal increments dB τ and dτ , respectively, as pIq :" ´pθ, qptu s T s λ q p1q λ pwq ¯1 0 "

ż T 0 " D Č t Ý Ñ F τ,T s λ ptu τ s λ , wq
‰ ppIq 1 pτ, wqdB τ `pIq 2 pτ, wqdτ q .

Then it is standard to use Burkholder's inequality and Hölder's inequality to deduce that

2 1´p E › › ›pIq › › › p ď ˜E ˇˇˇˇż T 0 › › › › " D Č t Ý Ñ F τ,T s λ ptu τ s λ , wq ‰ pIq 1 pτ, wq › › › › 2 dτ ˇˇˇˇq ¸1 2 `E › › › › ż T 0 " D Č t Ý Ñ F τ,T s λ ptu τ s λ , wq ‰ pIq 2 pτ, wq dτ › › › › q .
Using Corollary 5.33 and (4.44), we can continue to estimate E }pIq 1 } 4q , E }pIq 2 } 2q as in Proposition 5.29 and show they have same bound type in (5.2) with l " 0.

To complete the proof of i), we apply Lemma 5.5. It remains to show pp λ q p1q λ pT, x, yq is continuous in the pT, yq-coordinate, locally uniformly in λ, which is true if we have 1) the continuity of y Þ Ñ pp λ q p1q λ pT, x, yq, locally uniformly in T and λ, and 2) the continuity of T Þ Ñ pp λ q p1q λ pT, x, yq for every x, y fixed, locally uniformly in λ.

For 1), it holds if the continuity of y Þ Ñ pln p λ q p1q λ pT, x, yq is locally uniform in T and λ, where the latter is true if y Þ Ñ pΦ 1 λ pY qpF s yqq 1 0 is continuous, locally uniformly in T and λ. Since all the bounds in Lemmas 5.26-5.28 are locally uniform in py, T q and λ, the limits for continuity of y Þ Ñ pΦ 1 λ pY qpF s yqq 1 0 in proof of Lemma 5.34 are all locally uniform in T and λ.

We proceed to show 2). Simply denote by px λ , u λ q the stochastic pair which defines the Brownian motion starting from x. Then for any smooth function f on Ă M with support contained in a small neighborhood of y,

f px λ T q " f pxq `ż T 0 ∆ λ f px λ t q dt `ż T 0
H λ pu λ t , e i q `r f pu λ t q ˘dB i t .

Taking expectations on both sides shows

E `f px λ T q ˘" ż T 0 E `∆λ f px λ t q ˘dt.
Hence for T 1 ą T ,

E `f px λ T 1 q ˘´E `f px λ T q ˘" ż T 1 T E `∆λ f px λ t q ˘dt.
Differentiating both sides in λ gives ż

Ă M f pzq ´pp λ q p1q λ pT 1 , x, zq ´pp λ q p1q λ pT, x, zq ¯dVol λ pzq " ´ż Ă M f pzq ´pλ pT 1 , x, zq ´pλ pT, x, zq ¯pln ρ λ q p1q λ pzq dVol λ pzq `ż T 1 T E ´`∆ λ f ˘p1q λ px λ t q ¯dt `ż T 1 T ż Ă M `∆λ f ˘pzqφ 1 λ pt, x, zqp λ pt, x, zq dVol λ pzq,
where, as T 1 Ñ T , the first term tends to zero since p λ pT, x, zq is continuous at T ą 0, locally uniformly in z, the second term tends to zero since E ``∆ λ f ˘p1q λ px λ t q ˘is uniformly bounded for t in a small neighborhood of T and the last term goes to zero as well by using that the bound in (5.2) with l " 0 is locally uniform in t. In summary, we have lim

T 1 ÑT ż Ă M
f pzq ´pp λ q p1q λ pT 1 , x, zq ´pp λ q p1q λ pT, x, zq ¯dVol λ pzq " 0.

Since z Þ Ñ pp λ q p1q λ pT, x, zq is continuous, locally uniformly in T and λ, and f is arbitrary, we must have lim T 1 ÑT pp λ q p1q λ pT 1 , x, yq " pp λ q p1q λ pT, x, yq, locally uniformly in λ. This shows 2).

Finally, we show iii). By symmetry, the mapping x Þ Ñ pp λ q p1q λ pT, x, yq is continuous for all T, y, locally uniformly in y. Therefore iii) holds for any bounded function with compact support. Fix q ě 1. Any uniformly continuous and bounded r f P Cp Ă M q can be approximated by a sequence t r f n u nPN of continuous functions on Ă M with compact support in such a way that (5.97) lim

nÑ8 › › › r f pyq ´r f n pyq › › › q " 0,
locally uniformly in x. Property iii) follows by using (5.97) and (5.2) with l " 0.

Proof of Theorem 5.1 (k ą 3). By Theorem 5.1 i) of the k " 3 case and Lemma 5.5, we deduce Theorem 5.1 i). Hence ∇ plq pln p λ q p1q λ pT, x, ¨q, l ď k ´3, are well-defined. By taking the gradients of the identity (5.1), we obtain that ∇ plq φ 1 λ pT, x, ¨q, l ď k ´3, exist as well. For (5.2), it suffices to show the same type of L q -norm bounds hold for ∇ plq φ 1 λ pT, x, ¨q, l ď k ´3.

The l " 0 case was treated in the previous proof of Theorem 5.1 with k " 3. We proceed to consider the l " 1 case. Let W be a smooth bounded vector field on Ă M and let t r F u rPR be the flow it generates. Then

∇ λ W pyq φ 1 λ pT, x, ¨q " d dr ˇˇˇr "0 `φ1 λ pT, x, r F pyqq ˘.
We will look for a conditional expectation expression of ∇ λ W pyq φ 1 λ pT, x, ¨q and use it to estimate |∇φ 1 λ pT, x, ¨q|. For this, we adopt the idea we used in analyzing the regularity of Φ 1 λ pY q (see Section 5.1). Let f be an arbitrary bounded measurable function on Ă M . By the definition of the conditional expectation and the change of variable formula under r F ,

E ´r φ 1 λ pT, x, wqf ptx T s λ pwqq ¯" E ´E`r φ 1 λ pT, x, wq ˇˇtx T s λ pwq " y ˘f pyq " ż φ 1 λ pT, x, yqf pyqp λ pT, x, yqdVol λ pyq " ż φ 1 λ pT, x, r F pyqqf p r F pyqqp λ pT, x, r F pyqqdVol λ p r F pyqq.
Let t r Fs λ be the extension of r F to C x pr0, T s, Ă M q constructed in the previous subsections. By Proposition 5.23, all probabilities P λ x ˝tr Fs λ are absolutely continuous with respect to P λ x . Hence, using the change of variable formula under t r Fs λ , we obtain

E ´r φ 1 λ ¨f ptx T s λ q ¯" E ˜r φ 1 λ ˝tr Fs λ ¨f ˝tr Fs λ ¨dP λ x ˝tr Fs λ dP λ x " ż E ˜r φ 1 λ ˝tr Fs λ ¨dP λ x ˝tr Fs λ dP λ x ˇˇˇˇt x T s λ " y ¸f p r F pyqqp λ pT, x, yqdVol λ pyq.
Since f is arbitrary, a comparison of the two expressions of E `r φ 1 λ ¨f ptx T s λ q ˘shows φ 1 λ pT, x, r F pyqq " E ˜r φ ˝tr Fs λ q 1 r conditioned on x T " y are L 2 integrable, locally uniformly in the r parameter, we are allowed to take the differentiation under the expectation sign:

˜E˜r φ 1 λ ˝tr Fs λ ¨dP λ x ˝tr Fs λ dP λ x ˇˇˇˇt x T s λ " y ¸¸1 0 " E ˜˜r φ 1 λ ˝tr Fs λ ¨dP λ x ˝tr Fs λ dP λ x ¸1 0 ˇˇˇˇt x T s λ " y " E ´`r φ 1 λ ˝tr Fs λ ˘1 0 `r φ 1 λ ¨0E T ˇˇtx T s λ " y ¯.
Altogether, we will have ∇ λ W pyq φ 1 λ pT, x, ¨q " φ 1 λ pT, x, yq ´∇λ W pyq pln p λ pT, x, ¨qq `∇λ W pyq pln ρ λ q Ē

´`r φ 1 λ ˝tr Fs λ ˘1 0 `r φ 1 λ ¨0E T ˇˇtx T s λ " y (5.98)

and we can use it to show that a L q -norm bound as in (5.2) is valid for ∇ λ φ 1 λ pT, x, ¨q. We show ‹q first. Consider the processes t r τ s λ :" t τ s λ ˝tr Fs λ , " Dt r F τ,t s λ pt r τ s λ , wq ‰ ´1 :" " DtF τ,t s λ pt τ s λ , wq ‰ ´1 ˝tr Fs λ .

They are well-defined by Theorem 5.17 and the corresponding estimations in Lemmas 5.26-5.28 (for t r Fs λ ) are valid. Note that Dπptu T s λ q p1q λ , ∇ λ T,V,s Dπptu T s λ q p1q λ can be expressed by stochastic integrals using t τ s λ and tFs λ (see Proposition 5.29 and Proposition 5.32). Their images under t r Fs λ can be defined by applying t r Fs λ to each components in the integrals. So r φ 1 λ ˝tr Fs λ is well-defined. By using Lemma 4.13, Proposition 4.27 ii) and (5.96), it is easy to obtain

E P λ,x ,y,T ˇˇr φ 1 λ ˝tr Fs λ ˇˇ2 ď c ˆ`1 T d r g λ px, yq `1 ? T ˘2 `1˙e cp1`d r g λ px,yqq
for some constants c (depending on s, r 0 , }g λ } C 3 and }X λ } C 2 ) and c (depending on T, T 0 and }g λ } C 3 ). By Propositions 4.14, 5.20 and 5.23, we may also assume c, c are such that

`EP λ,x ,y,T ˇˇ0 E T ˇˇ2 ˘1 2 ď }W pyq}ce cp1`d r g λ px,yqq .
To justify (5.98), it remains to check the differentiability of r Þ Ñ A ˝tr Fs λ , for A " pIq, pIIq, pIIIq, pIVq in (5.96) and show the differentials `A ˝tr Fs λ ˘1 r are L 2 integrable, uniformly in the r parameter. We begin with A " pIVq. By Proposition 5.30, pDπptu T s λ q p1q λ q tr Fs λ is differentiable in r. Let r P r´r 0 , r 0 s. As usual, we write t r xs λ :" txs λ ˝tr Fs λ , t r us λ :" tus λ ˝tr Fs λ , ∇ r,λ T,W,s Dπptu T s λ q p1q λ :" ´pDπpt r u T s λ q p1q λ q ¯1 r .

Then pIVq ˝tr Fs λ is differentiable in r with differential

`pIVq ˝tr Fs λ ˘1 r " ´@∇ r,λ T,W,s Dπptu T s λ q p1q λ , ∇ λ ln p λ pT, x, t r x T s λ q D λ ´@Dπpt r u T s λ q p1q λ , ∇ λ Wpt r x T s λ q ∇ λ ln p λ pT, x, t r x T s λ q D λ .
By Proposition 4.27, we can obtain some c 1 (depending on s, r 0 , }g λ } C 3 and }X λ } C 1 ) and c 1 (depending on T, T 0 and }g λ } C 3 ) such that

ˆEP λ,x ,y,T › › Dπpt r u T s λ q p1q λ › › 2 ˙1 2 ď }W pyq}c 1 e c 1 p1`d r g λ px,yqq .
By Proposition 5.29, we can obtain some c 2 (depending on s, r 0 , }g λ } C 3 and }X λ } C 2 ) and c 2 (depending on T, T 0 and }g λ } C 3 ) such that

ˆEP λ,x ,y,T › › ∇ r,λ T,W,s Dπptu T s λ q p1q λ › › 2 ˙1 2 ď }W pyq}c 2 e c 2 p1`d r g λ px,yqq .
Using (4.33), we further obtain

ˆEP λ,x ,y,T ˇˇ`p IVq ˝tr Fs λ ˘1 r ˇˇ2 ˙1 2 ď ˆEP λ,x ,y,T › › ∇ r,λ T,W,s Dπptu T s λ q p1q λ › › 2 ˙1 2 › › ∇ λ ln p λ pT, x, t r F s λ pyqq › › `ˆE P λ,x ,y,T › › Dπpt r u T s λ q p1q λ › › 2 ˙1 2 › › ∇ λ W pt r F s λ pyqq ∇ λ ln p λ pT, x, t r F s λ pyqq › › ď }W pyq}c 3 e c 3 p1`d r g λ px,yqq 2 ÿ i"1 `1 T d r g λ px, t r F s λ pyqq `1 ? T ˘i,
where c 3 (depending on s, r 0 , }g λ } C 3 and }X λ } C 2 ) and c 3 (depending on T, T 0 and }g λ } C 3 ) and this bound is finite and is uniform in r. For A " pIIq, pIIIq, the same argument shows the C 1 regularity of r Þ Ñ A ˝tr Fs λ and

ˆEP λ,x ,y,T ˇˇ`A ˝tr Fs λ ˘1 r ˇˇ2 ˙1 2 ď }W pyq}c A e c A p1`d r g λ px,yqq
for some c A (depending on s, r 0 , }g λ } C 3 and }X λ } C 1 ) and c A (depending on T, T 0 and }g λ } C 3 ). It remains to analyze pIq ˝tr Fs λ . Recall that for any smooth bounded vector field V on Ă M ,

∇ λ T,V,s Dπptu T s λ q p1q λ " Dπ `ptu s T s λ q p1q λ pwq ˘1 0 ,
where `ptu s T s λ q p1q λ pwq ˘1 0 was formulated in (5.87). Hence the regularity of r Þ Ñ pIq˝t r Fs λ can be reduced to the regularity of each component of (5.87) under t r Fs λ . Applying Theorem 5.17 to t r Fs λ shows r Þ Ñ t r τ s λ , " Dt r F τ,t s λ pt r τ s λ , wq ‰ ´1 are C 1 . Lemmas 5.26-5.28 also hold true for t r Fs λ . Using these properties and the fact that λ Þ Ñ g λ is C k in M k pM q with k ě 4, we can deduce the regularity of the components of (5.87) under t r Fs λ . Moreover, by a routine computation using Lemmas 5.26-5.28, we can obtain some c I (depending on s, r 0 , }g λ } C 4 and }X λ } C 3 ) and c I (depending on T, T 0 and }g λ } C 3 ) such that

ˆEP λ,x ,y,T ˇˇ`pIq ˝tr Fs λ ˘1 r ˇˇ2 ˙1 2 ď }W pyq}c I e c I p1`d r g λ px,yqq .
Altogether, we have the differentiability of λ Þ Ñ r φ 1 λ ˝tr Fs λ and also obtain some c (depending on s, r 0 , }g λ } C 4 and }X λ } C 3 ) and c (depending on T, T 0 and }g λ } C 3 ) such that

ˆEP λ,x ,y,T ˇˇ`r φ 1 λ ˝tr Fs λ ˘1 0 ˇˇ2 ˙1 2 ď }W pyq}ce cp1`d r g λ px,yqq ˆp 1 T d r g λ px, yq `1 ? T q 2 `1˙.
Now (5.98) holds true. Using Hölder's inequality, it is easy to deduce

4 1´q › › ›∇ λ W pyq φ 1 λ pT, x, ¨q› › › q L q ď › › φ 1 λ pT, x, ¨q› › q L 2q ´› › ›∇ λ W pyq pln p λ pT, x, ¨qq › › › q L 2q `› › ›∇ λ W pyq pln ρ λ q › › › q L 2q Ē› › r φ 1 λ › › 2q ˘1 2 `E › › 0 E T › › 2q ˘1 2 `E› › `r φ 1 λ ˝tr Fs λ ˘1 0 › › q ": D 1 pqq `D2 pqq `D3 pqq.
By (4.33), (4.31), we see that, for the i-th covariant derivative ∇ λ,piq ln p λ pt, x, ¨q, i ď k ´2, there is cpiq (depending on m, q, T 0 and }g λ } C i`2 ) and cpiq (depending on }g λ } C 3 ) such that (5.99) › › ›∇ λ,piq ln p λ pT, x, ¨q› › › q L q ď cpiqe cpiqp1`T q . Using this and the L q estimation of φ 1 λ pT, x, ¨q in the proof of Theorem 5.1 for the k " 3 case, we obtain D 1 pqq ď c λ pqq, where c λ pqq depends on m, q, T 0 , T , }g λ } C 3 and }X λ } C 2 . With the L 2q estimations of r φ 1 λ pT, x, ¨q and 0 E T for Theorem 5.1 with k " 3, we can also conclude that D 2 pqq has the same type of bound as D 1 pqq. For D 3 pqq, we check the L qnorms of `A ˝tr Fs λ ˘1 0 for A " pIq, pIIq, pIIIq or pIVq in (5.96), respectively. Using Hölder's inequality, (5.99) and Proposition 4.27, it suffices to estimate the L q -norms of ´pt r u T s λ q p1q λ ¯1 r"0 , ˆ`ptu s T s λ q p1q λ pwq ˘1 0 ˝tr Fs λ ˙1 r"0 .

This, by using Lemma 5.31 and Proposition 5.32, can be eventually reduced to a multiple of a constant depending on m, q, T 0 , T , }g λ } C 4 and }X λ } C 3 with a combination of some L q 1 norm estimations (with q 1 ě 1 depending on q) of sup 0ďtďT }ptu t s λ q p1q λ } and sup

0ďtătďT › › " D Č t Ý Ñ F t,t s λ ptu t s λ , wq ‰› › .
Hence, by Proposition 4.27, we conclude that D 3 pqq has the same type of bound as D 1 pqq with c λ pqq depending on m, q, T 0 , T , }g λ } C 4 and }X λ } C 3 .

For the L q -norm estimation of ∇ p2q φ 1 λ pT, x, ¨q, we continue to differentiate (5.98). Let W 2 be another smooth bounded vector field on Ă M . Then

∇ λ W 2 pyq ∇ λ W pyq φ 1 λ pT, x, ¨q "φ 1 λ pT, x, ¨q ´∇λ W 2 pyq ∇ λ W pyq pln p λ pT, x, ¨qq `∇λ W 2 pyq ∇ λ W pyq pln ρ λ q ∇λ W 2 pyq φ 1 λ pT, x, ¨q ´∇λ W pyq pln p λ pT, x, ¨qq `∇λ W pyq pln ρ λ q ∇λ W 2 pyq ´E ´`r φ 1 λ ˝tr Fs λ ˘1 0 `r φ 1 λ ¨0E T ˇˇtx T s λ " y ¯":paq y `pbq y `pcq y .
Using the previous estimations of φ 1 λ , ∇ λ W φ 1 λ , (5.99) and Hölder's inequality, we obtain

› › paq y › › L q , › › pbq y › › L q ď }W 2 pyq}}W pyq}c a,b e c a,b T ,
where c a,b depends on m, q, }g λ } C 4 , }X λ } C 3 and c a,b depends on m, q, T, T 0 and }g} C 3 . For pcq y , we can follow the above argument for ∇ λ W pyq φ 1 λ pT, x, ¨q to 'exchange' the differentiation ∇ λ W 2 pyq with the conditional expectation sign and obtain

pcq y "E ˆd da ˇˇˇa "0 ´``r φ 1 λ ˝tr Fs λ ˘1 0 `r φ 1 λ ¨0E T ˘˝t a F W 2 s λ ¯ˇˇˇt x T s λ " y Ė ´``r φ 1 λ ˝tr Fs λ ˘1 0 `r φ 1 λ ¨0E T ˘0E W 2 T ˇˇtx T s λ " y ¯, where t a F W 2 s λ , 0 E W 2
T are the corresponding objects t a Fs λ , 0 E T for W 2 . In addition to the terms involving a single differentiation of t a F W 2 s λ or t r F W 1 s λ , we have the differentiation of `r φ 1 λ ˝tr Fs λ ˘1 0 under t a F W 2 s λ , which involves ∇ λ W 2 pyq ∇ λ W pyq ∇ λ ln p λ pT, x, ¨q and multistochastic integrals using the tangent maps " DtF τ,t s λ pt τ s λ , wq ‰ and geometric terms with bounds determined by }g λ } C 5 and }X λ } C 4 . So, a routine calculation as above using Proposition 4.27 gives › › pcq y › › L q ď }W 2 pyq}}W pyq}c c e ccT , where c c depends on m, q, }g λ } C 5 and }X λ } C 4 , and c c depends on m, q, T, T 0 and }g} C 3 .

Continuing this argument, we can obtain the estimations in (5.2) for all l ď k ´3. We stop at l " k ´3 step since ∇ plq φ 1 λ pT, x, ¨q involves ∇ pl`1q pln p λ qpT, x, ¨q and the bound estimation in (4.33) is only valid for ∇ plq pln p λ qpT, x, ¨q, l ď k ´2, in general.

In proving (1.4), we also obtain the following coarse estimation, which will be used in the inductive argument in the next section.

Corollary 5.36. For all l, 0 ď l ď k ´3, there is c λ,pl,1q , depending on m, }g λ } C l`3 and }X λ } C l`2 , and c λ,pl,1q , depending on l, m, q, T, T 0 and }g λ } C 3 , such that ˇˇ∇ plq pln p λ q p1q λ pT, x, yq ˇˇ, ˇˇ∇ plq φ 1 λ pT, x, yq ˇď pp λ pT, x, yqq ´1c λ,pl,1q pl,1q p1`d r g λ px,yqq .

ˆ`1 T d r g λ px, yq `1 ? T ˘l`1 `1˙¨e c λ,

Higher order regularity of the heat kernels in metrics

To conclude Theorem 1.3 for all i, 2 ď i ď k ´2, we use an inductive argument based on the proof of Theorem 5.1 to identify the differentials pp λ q piq λ pT, x, ¨q, 2 ď i ď k ´2, using the SDE theory in Section 4. The estimations in (1.4) and (1.5) will be obtained using the conditional stochastic expressions of tpln p λ q piq λ pT, x, ¨qu. In the following, we first pick out the properties of pp λ q piq λ pT, x, ¨q necessary for an inductive argument, then verify these properties for the i " 2 case and the i ą 2 case, respectively. f pyqφ i λ pT, x, yqp λ pt, x, yq dVol λ pyq.

Proof. Assume (6.1) holds true. We show the differentials pp λ q piq λ pT, x, ¨q, i " 1, ¨¨¨, k ´2, exist as continuous functions on Ă M and satisfy (6.2) j ÿ i"0 ˆj i ˙pp λ q piq λ pT, x, yqpρ λ q pj´iq pyq " φ j λ pT, x, yqp λ pT, x, yqρ λ pyq, j " 1, ¨¨¨, k ´2.

The j " 1 case was handled in Lemma 5.5 and we know that pp λ q p1q λ pT, x, ¨q is a continuous function on R `ˆĂ M . Assume pp λ q piq λ pT, x, ¨q, i ď j 0 ă k ´2, exist, are continuous, and satisfy (6.2) for j ď j 0 . Using this, a comparison of (6.1) for i " j 0 and j 0 `1 gives ż Ă M ´φj 0 λ pT, x, yqp λ pT, x, yqρ λ pyq ´φj 0 0 pT, x, yqp 0 pT, x, yqρ 0 pyq ¯f pyq dVol 0 pyq "

ż Ă M ˆż λ 0 φ j 0 `1 r λ pT, x, yqp r λ pT, x, yqρ r λ pyqd r λ ˙f pyq dVol 0 pyq, @f P C 8 c p Ă M q.
Since both sides are continuous functions in y-variable, we must have φ j 0 λ pT, x, yqp λ pT, x, yqρ λ pyq ´φj 0 0 pT, x, yqp 0 pT, x, yqρ 0 pyq "

ż λ 0 φ j 0 `1 r λ pT, x, yqp r λ pT, x, yqρ r λ pyqd r λ.
Consequently,

˜j0 ÿ i"0 ˆj0 i ˙pp λ q piq λ pT, x, yqpρ λ q pj 0 ´iq λ pyq ¸p1q λ " φ j 0
`1 λ pT, x, yqp λ pT, x, yqρ λ pyq, which implies that pp λ q pj 0 `1q λ pT, x, yq exists for every y and satisfies (6.2). Then we can conclude from this and the inductive assumption on the continuity of pp λ q piq λ pT, x, ¨q, i " 1, ¨¨¨, j 0 , that pp λ q pj 0 `1q λ pT, x, ¨q is also a continuous function on R `ˆĂ M . Now, the differentials pp λ q piq λ pT, x, ¨q, i " 1, ¨¨¨, k ´2, exist as continuous functions on R `ˆĂ M and hence their weak derivatives in pT, yq of any order are well-defined. Taking the differential of the heat equations L λ p λ " 0 in λ gives the following identities in distribution: L λ,w pp λ q piq λ pT, x, ¨q `i ÿ j"1 ˆi j ˙pL λ q pjq,w pp λ q pi´jq λ pT, x, ¨q " 0, i " 1, ¨¨¨, k ´2, where pL λ q pjq,w is the weak derivative of the j-th differential operator pL λ q pjq λ . We can use Lemma 5.4 and Lemma 5.2 inductively to improve the regularity of pp λ q piq λ pT, x, ¨q. Shrinking the neighborhood V g of g if necessary, we may assume there is ι ą 0 such that p λ pT, x, ¨q P C k,ι p Ă M q for all λ. Since it is a local problem, for pT, yq P R `ˆĂ M , we can also restrict ourselves to a bounded domain D containing pT, yq. By Lemma 5.5, there is some domain D 1 Ă D such that p λ pT, x, ¨q, pp λ q p1q λ pT, x, ¨q P C k,ι pD 1 q. Assume for all i ď j 0 ă k ´2 there are domains D i containing pT, yq such that |pp λ q piq λ pT, x, ¨q| 0,2`ι ă 8 on D i and pp λ q piq λ pT, x, ¨q P C k,ι pD i q. Then L λ,w pp λ q pj 0 `1q λ pT, x, ¨q " ´j0 `1

ÿ j"1 ˆj0 `1 j ˙pL λ q pjq,w pp λ q pj 0 `1´jq λ pT, x, ¨q " ´j0 `1 ÿ j"1 ˆj0 `1 j ˙pL λ q pjq λ pp λ q pj 0 `1´jq λ
pT, x, ¨q. (6.3) Shrinking D j 0 to D j 0 `1 if necessary, we can deduce from |pp λ q piq λ pT, x, ¨q| 0,2`ι ă 8 on D i that |pL λ q pjq pp λ q pj 0 `1´jq λ pT, x, ¨q| 2,ι is finite for all j ď j 0 `1 on D j 0 `1. Since pp λ q pj 0 `1q λ pT, x, ¨q is continuous, Lemma 5.4 shows that (6.3) holds in the usual sense. Then we can apply Lemma 5.3 to conclude that |pp λ q pj 0 `1q λ pT, x, ¨q| 0,2`ι ă 8 on D j 0 `1 and apply Lemma 5.2 to conclude pp λ q pj 0 `1q λ pT, x, ¨q P C k,ι pD j 0 `1q. Accordingly, the continuity of λ Þ Ñ pp λ q p1q λ pT, x, ¨q in Cp Ă M q can be improved to be the continuity in C k,ι p Ă M q by using the parabolic differential equation (6.3), Lemma 5.2 and Lemma 5.3.

The φ 1 λ satisfying (6.1) was identified in Theorem 5.1. We continue to pick up a candidate φ 2 λ for (6.1). Let r φ 1 λ pT, x, ¨q be as in (5.96) such that (5.95) holds. Then, for any

f P C 8 c p Ă M q, ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙p1q λ " E ´f ptx T s λ pwqq r φ 1 λ pT, x, wq ¯. (6.4) If we can show λ Þ Ñ r φ 1
λ is differentiable, and both r φ 1 λ and the differential p r φ 1 λ q p1q λ are L q integrable for some q ě 1, we are allowed to differentiate under the expectation sign of the right hand side term of (6.4). This will give

ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙p2q λ " ż Ă M f pyqE ´p r φ 1 λ q p1q λ pT, x, wq ˇˇtx T s λ pwq " y ¯pλ pT, x, yq dVol λ pyq `E ´@∇ λ tx T s λ pwq f ptx T s λ pwqq, r φ 1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq D λ ¯. (6.5)
We can deal with the last expectation term in (6.5) as we did for φ 1 λ in Section 5. Define Φ 2 λ pY, wq :"

@ Y ptx T s λ pwqq, r φ 1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq D λ ,
where Y is any C k bounded vector field on Ă M , and consider the linear functional

Φ 2 λ : Y Þ Ñ E ´Φ2 λ pY, wq ˇˇtx T s λ pwq " y ¯. If we can show Φ 2 λ is such that Φ 2 λ pY q is C 1 in y variable, we can conclude that (6.6) E ´r φ 1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq ˇˇtx T s λ pwq " y ¯": z λ,2 T pyq is a C 1 vector field on Ă M and satisfies Φ 2 λ p∇f qpyq " @ ∇ λ y f pyq, z λ,2 T pyq D λ .
Using the classical integration by parts formula, we obtain

E ´@∇ λ tx T s λ pwq f ptx T s λ pwqq, r φ 1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq D λ " ż Ă M @ ∇ λ y f pyq, p λ pT, x, yqz λ,2 T pyq D λ dVol λ pyq " ´ż Ă M f pyq ´Div λ z λ,2
T pyq `@z λ,2 T pyq, ∇ λ ln p λ pT, x, yq D λ ¯pλ pT, x, yq dVol λ pyq.

Therefore, a candidate of φ 2 λ pT, x, ¨q for (6.1) is

φ 2 λ pT, x, yq :" E ´p r φ 1 λ q p1q λ pT, x, wq ˇˇtx T s λ pwq " y Div λ z λ,2 T pyq `@z λ,2 T pyq, ∇ λ ln p λ pT, x, yq D λ ¯. (6.7)
Once we show φ 2 λ pT, x, yq fulfills the continuity requirement of Lemma 6.1, we can conclude the second order differentiability of λ Þ Ñ p λ pT, x, q in C k,ι for some ι ą 0. It follows that pln p λ q p2q λ pT, x, yq " φ 2 λ pT, x, yq ´pφ 1 λ q 2 pT, x, yq ´pln ρ λ q p2q λ pyq. Note that the gradients estimations of φ 1 λ were already handled in Theorem 5.1. Hence the gradients estimations of pln p λ q p2q λ can be reduced to that of φ 2 λ , which can be analyzed following the proof of Theorem 5.1, if we can find some controllable r φ 2 λ pT, x, ¨q such that φ 2 λ pT, x, yq " E ´r φ 2 λ pT, x, wq ˇˇtx T s λ pwq " y ¯.

We will follow this line of discussion to find all the candidates φ i λ for (6.1). Put r φ 0 λ pT, x, wq " 1 and let r φ 1 λ pT, x, wq be as in (5.96). For i, 2 ď i ď k ´2, define r φ i λ pT, x, wq :" `r φ i´1 λ pT, x, wq

˘p1q λ ´@∇ λ T,s r φ i´1 λ pT, x, wq, Dπptu T s λ q p1q λ pwq D (6.8) `r φ i´1 λ pT, x, wq r φ 1 λ pT, x, wq,
where the 'path-wise gradient' ∇ λ T,s r φ i´1 λ pT, x, wq will be specified later. We will show each φ i λ pT, x, yq :" E ´r φ i λ pT, x, wq ˇˇtx T s λ pwq " y (6.9) fulfills all the requirements in Lemma 6.1. The stochastic expression (6.9) will be used for two purposes: one is for the gradient estimations of φ i λ pT, x, ¨q and pln p λ q piq λ pT, x, ¨q; the other is for obtaining φ i`1 λ pT, x, yq as we exposed above for φ 2 λ pT, x, yq (see (6.7)). Let us highlight the necessary steps to undergo an inductive argument for Theorem 1.3. Assume for all i ă j ď k ´2, the φ i λ defined in (6.9) are such that Lemma 6.1 holds true, pp λ q piq λ pT, x, ¨q P C k,ι p Ă M q, is continuous in λ and (1.4) holds for pln p λ q piq λ pT, x, ¨q. We also assume the following coarse pointwise estimation holds true for all i ă j. 0) For all l, 0 ď l ď k ´2 ´i, there is c λ,pl,iq depending on }g λ } C l`i`2 , }X λ } C l`i`1 and c λ,pl,iq depending on pl, iq, m, q, T, T 0 and }g λ } C 3 such that ˇˇ∇ plq pln p λ q piq λ pT, x, yq ˇˇ, ˇˇ∇ plq φ i λ pT, x, yq ˇď pp λ pT, x, yqq ´ic λ,pl,iq ˆ`1 T d r g λ px, yq `1 ? T ˘l`i `1˙¨e c λ,pl,iq p1`d r g λ px,yqq . (6.10) For the existence of pp λ q pjq λ pT, x, ¨q, the very first step is find some measurable candidate satisfying (6.1), which can be done once we show the following.

i) The function r φ j´1 λ pT, x, wq is differentiable in λ for almost all w P Θ `and both r φ j´1 λ pT, x, wq and `r φ j´1 λ ˘p1q λ pT, x, wq are L q integrable in w for all q ě 1. ii) For any C k bounded vector field Y on Ă M , let Φ j λ pY, wq :"

@ Y ptx T s λ pwqq, r φ j´1 λ pT, x, wqDπptu T s λ q p1q λ pwq D λ . Then the linear functional Φ j λ : Y Þ Ñ E ´Φj
λ pY, wq ˇˇtx T s λ pwq " y īs bounded with Φ j λ pY qpyq varying C 1 in the y-coordinate. Claim 6.2. Assume i), ii) are true. Then (6.1) hold with some φ λ pT, x, ¨q for i " j.

Proof. By the inductive assumption, ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙pj´1q λ " E ´f ptx T s λ q r φ j´1 λ pT, x, wq ¯. (6.11) If i) is true, we can differentiate under the expectation sign of (6.11). This gives

ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙pjq λ " E ´f ptx T s λ pwqqp r φ j´1 λ q p1q λ pT, x, wq ¯`E ´Φj λ p∇f, wq ¯.
The property ii) implies (6.12) z λ,j T pyq :" E ´r φ j´1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq ˇˇtx T s λ pwq " y īs

a C 1 vector field on Ă M such that Φ j λ pY qpyq " @ Y pyq, z λ,j T pyq D λ .
In particular, we have

E ´Φj λ p∇f, wq ¯" ż Ă M @ ∇ λ y f pyq, p λ pT, x, yqz λ,j T pyq D λ dVol λ pyq " ´ż Ă M f pyq ´Div λ z λ,j T pyq`@z λ,j T pyq, ∇ λ ln p λ pT, x, yq D λ ¯pλ pT, x, yq dVol λ pyq.
This means a measurable candidate φ j λ for (6.1) at i " j is φ λ pT, x, yq :" E ´p r φ j´1 λ q p1q λ pT, x, wq ˇˇtx T s λ pwq " y Div λ z λ,j T pyq `@z λ,j T pyq, ∇ λ ln p λ pT, x, yq D λ ¯. (6.13)

We will show φ λ pT, x, ¨q given in Claim 6.2 coincides with φ j λ pT, x, ¨q defined by (6.9). For any smooth bounded vector field V on Ă M , let tF s u sPR be the flow it generates. As we did for Φ 1 λ in Section 5 (see Lemma 5.34), we will prove the following in verifying ii).

iii) For any y P Ă M , s Þ Ñ Φ j λ pY qpF s yq is differentiable at s " 0 and the differential pΦ j λ pY qpF s yqq 1 0 varies continuously in y. Moreover,

pΦ j λ pY qpF s yqq 1 0 " Φ j λ p∇ λ V Y qpyq `E ´@Y ptx T s λ pwqq, ∇ λ T,V,s `r φ j´1 λ pT, x, wqDπptu T s λ q p1q λ pwq ˘Dλ `@Y ptx T s λ pwqq, `r φ j´1 λ pT, x, wqDπptu T s λ q p1q λ pwq ˘Dλ Ð Ý E T,V,s pwq ˇˇtx T s λ pwq " y ¯,
where the path-wise differential ∇ λ T,V,s `r φ i´1 λ pT, x, wqDπptu T s λ q p1q λ pwq ˘will be clarified later and it satisfies ∇ λ T,V,s `r φ i´1 λ pT, x, wqDπptu T s λ q p1q λ pwq

˘" `∇λ T,V,s r φ i´1 λ pT, x, wq ˘¨Dπptu T s λ q p1q λ pwq `r φ i´1 λ pT, x, wq ¨`∇ λ T,V,s Dπptu T s λ q p1q λ pwq ˘.

Claim 6.3. Assume i)-iii) are true. Then φ λ pT, x, ¨q " φ j λ pT, x, ¨q.

Proof. By ii), both Φ j λ pY qpyq and z λ,j T pyq vary C 1 in y. Hence

pΦ j λ pY qpF s yqq 1 0 " ∇ λ V pΦ j λ pY qqpyq " @ ∇ λ V Y pyq, z λ,j T pyq D λ `@Y pyq, ∇ λ V z λ,j T pyq D λ .
Comparing this with the expression of `Φj λ pY qpF s yq ˘1 0 in iii) gives

∇ λ Vpyq z λ,j T pyq " E ´∇λ T,V,s `r φ j´1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq ȓ φ j´1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq
˘Ð Ý E T,V,s pwq ˇˇtx T s λ pwq " y ¯. (6.14) Following the argument in the proof of Lemma 5.35 and then using (5.96), we obtain Div λ z λ,j T pyq

" E ´tr `V Þ Ñ ∇ λ T,V,s `r φ j´1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq ˘ȓ φ j´1 λ pT, x, wq ¨@Dπptu T s λ q p1q λ pwq, 1 2 ż T 0 spT ´τ qtu T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d Ð Ý B τ D λ `r φ j´1 λ pT, x, wq ¨@Dπptu T s λ q p1q λ pwq, ´1 2 tu T s λ ż T 0 s 1 pT ´τ qd Ð Ý B τ D λ ˇˇˇt x T s λ pwq " y " E ´@∇ λ T,s r φ j´1 λ pT, x, wq, Dπptu T s λ q p1q λ pwq D ˇˇtx T s λ pwq " y Ē ´r φ j´1 λ pT, x, wq `r φ 1 λ pT, x, wq`@Dπptu T s λ q p1q λ , ∇ λ ln p λ pT, x, tx T s λ q D λ ˘ˇˇt x T s λ pwq " y ¯.
Note that xz λ,j T pyq, ∇ λ ln p λ pT, x, yqy λ " E ´r φ j´1 λ pT, x, wq

@ Dπptu T s λ q p1q λ pwq, ∇ λ ln p λ pT, x, tx T s λ pwqq D λ ˇˇtx T s λ pwq " y ¯.
Reporting these two expressions in (6.13), we obtain

φ λ pT, x, yq " E ´`r φ j´1 λ pT, x, wq ˘p1q λ ´@∇ λ T,s r φ j´1 λ pT, x, wq, Dπptu T s λ q p1q λ pwq D `r φ j´1 λ pT, x, wq r φ 1 λ pT, x, wq ˇˇtx T s λ pwq " y " φ j λ pT, x, yq.
To study the continuity of φ j λ pT, x, yq in pT, yq, we first show the following.

iv) For all x P Ă M , T P R `, y Þ Ñ E ´p r φ j´1 λ q p1q λ pT, x, wq ˇˇtx T s λ pwq " y īs continuous, locally uniformly in T and λ. Moreover, there exist c λ,j (depending on }g λ } C j`2 , }X λ } C j`1 ) and r c λ,j (depending on j, T, T 0 and }g λ } C 3 ) such that

ˇˇˇE P λ x,y,T ´p r φ j´1 λ q p1q λ pT, x, wq ¯ˇˇď pp λ pT, x, yqq ´j c λ,j ˆ`1 T d r g λ px, yq `1 ? T ˘j `1˙¨e r c λ,j p1`d r g λ px,yqq .
v) For all x P Ă M , T P R `, the mappings y Þ Ñ z λ,j T pyq, y Þ Ñ Div λ z λ,j T pyq are continuous, locally uniformly in T and λ. Moreover, there are c 1 λ,j (depending on }g λ } C j`2 , }X λ } C j`1 ) and c λ,j depending on j, T, T 0 and }g λ } C 3 such that ˇˇz λ,j T pyq ˇˇ, ˇˇDiv λ z λ,j T pyq

ˇď pppT, x, yqq ´j c 1 λ,j ˆ`1 T d r g λ px, yq `1 ? T ˘j `1˙¨e c λ,j p1`d r g λ px,yqq .
vi) There is c λ,p0,jq pqq depending on j, m, q, T, T 0 , }g λ } C j`2 and }X λ } C j`1 such that › › ›φ j λ pT, x, ¨q› › › L q ď c λ,p0,jq pqq, @q ě 1.

(6.15) Claim 6.4. Assume i)-vi). For every x P Ă M , pT, yq Þ Ñ φ j λ pT, x, yq is continuous, locally uniformly in λ.

Proof. To conclude the continuity of φ j λ pT, x, yq in pT, yq, we verify the following. 1) For all x P Ă M , T P R `, y Þ Ñ φ j λ pT, x, yq is continuous, locally uniformly in T and λ. 2) For each x, y fixed, T Þ Ñ φ j λ pT, x, yq is continuous, locally uniformly in λ. By the inductive assumption, all pp λ q piq pT, x, yq, i ă j, exist, are continuous in pT, yq and satisfy the bound estimation (1.4). By i)-iii), φ j λ pT, x, yq " E ´p r φ j´1 λ q p1q λ pT, x, wq ˇˇtx T s λ pwq " y Div λ z λ,j T pyq `@z λ,j T pyq, ∇ λ ln p λ pT, x, yq D λ (6.16) satisfies (6.1). By iv) and v), for each x P Ă M , we have that the mapping y Þ Ñ φ j λ pT, x, yq is continuous, locally uniformly in T .

For 2), we follow the proof of Theorem 5.1 for the k " 3 case. Simply denote by px λ , u λ q the stochastic pair which defines the Brownian motion on p Ă M , r g λ q starting from x. Then, for any f P C 8 c p Ă M q with support contained in a small neighborhood of y and T 1 ą T ,

E `f px λ T 1 q ˘´E `f px λ T q ˘" ż T 1 T E `∆λ f px λ t q ˘dt.
Take the j-th differential in λ of both sides and use (6.1). We obtain ż

Ă M f pzq ´φj λ pT 1 , x, zq ´φj λ pT, x, zq ¯pλ pT 1 , x, zq dVol λ pzq `ż Ă M f pzqφ j λ pT, x, zq ´pλ pT 1 , x, zq ´pλ pT, x, zq ¯dVol λ pzq " ż T 1 T ż Ă M j ÿ i"0 ˆj i ˙p∆ λ f pzqq piq λ ´pλ pt, x, zqρ λ pzq ¯pj´iq λ dVol 0 pzq dt.
Using (6.15), we deduce that lim

T 1 ÑT ż Ă M
f pzq ´φj λ pT 1 , x, zq ´φj λ pT, x, zq ¯pλ pT 1 , x, zq dVol λ pzq " 0.

Since φ j λ pT, x, zq is continuous in z and λ, locally uniformly in T , and f is arbitrary, we must have lim T 1 ÑT φ j λ pT 1 , x, yq " φ j λ pT, x, yq, locally uniformly in λ. This shows 2) and finishes the proof of Claim 6.4. Claim 6.5. Assume i)-vi). Then for any x P Ă M , T P R `, λ Þ Ñ p λ pT, x, ¨q is C j in C k,ι p Ă M q for some ι ą 0. The differential pp λ q pjq λ pT, x, yq satisfies the equation (6.17) pp λ q pjq λ pT, x, yq " φ j λ pT, x, yqp λ pT, x, yq ´pρ λ pyqq ´1 j´1 ÿ i"0 ˆj i ˙pp λ q piq λ pT, x, yqpρ λ q pj´iq λ pyq.

Consequently, φ j λ pT, x, ¨q P C k,ι p Ă M q as well.

Proof. The function φ j λ pT, x, yq is continuous in y, uniformly in λ by using iv), v) and (6.16). So, it is continuous in λ if for any f P C 8 c p Ă M q, we have the continuity of λ Þ Ñ ˆż Ă M f pyqp λ pT, x, yq dVol λ pyq ˙pjq λ ": A j pλ, T, xq.

Note that

A 1 pλ, T, xq " E ´A∇ λ tx T s λ pwq pf ˝πqptu T s λ pwqq, ptu T s λ q p1q λ pwq

E λ ¯.
Differentiating A 1 pλ, T, xq in λ for j ´1 times, we get a similar expression A j pλ, T, xq of a combination of inner products involving t∇ piq f u iďj , tptx T s λ q piq u iďj and tptu T s λ q piq λ u iďj . Following Proposition 4.27 i), we can derive the L q pq ě 1q convergence of ptx T s λ q piq and ptu T s λ q piq λ in λ. As a consequence, we obtain the continuity of λ Þ Ñ A j pλ, T, xq. Now, by vi), the continuity of φ j λ pT, x, yq in λ and pT, yq and the induction assumption, we can apply Lemma 6.1 to conclude that λ Þ Ñ p λ pT, x, ¨q is C j in C k,ι p Ă M q for some ι ą 0. The equation (6.17) holds by comparison and hence φ j λ pT, x, ¨q P C k,ι p Ă M q.

With i)-vi), the gradients t∇ plq pln p λ q pjq λ pT, x, yqu 1ďlďk´2´j are well-defined. To conclude Theorem 1.3 ii) by induction, it remains to show (1.4) for i " j. With the identity (6.17) and vi), it remains to show the following.

vii) For all l, 1 ď l ď k ´2 ´j, q ě 1, there is c λ,pl,jq pqq which depends on pl, jq, m, q, T, T 0 , }g λ } C l`j`2 and }X λ } C l`j`1 such that › › ›∇ plq φ j λ pT, x, ¨q› › › L q ď c λ,pl,jq pqq.

(6.18) For (6.18), we will use (6.9) to formulate ∇ plq ∇ W 1 ,W 2 ,¨¨¨,W j φ 1 λ (for any smooth bounded vector fields W 1 , W 2 , ¨¨¨, W j ) as some conditional expectation and use it for evaluations as in the proof of Theorem 5.1. For this, we need the bounds control on φ j λ pT, x, yq from iv), v). Note that in showing iv), v), we need a bound control of t|∇ plq pln p λ q piq λ pT, x, ¨q|u 1ďlďj´i . So, to continue the inductive argument, we also need to verify 0) at i " j, which can be obtained in showing vi) and vii).

Theorem 1.3 iii) will follow from ii). Indeed, for i " 1, (1.5) is true by Theorem 5.1 for the k " 3 case. For i ě 2, by (6.17), pp λ q piq pT, x, yq p λ pT, x, yq " φ i λ pT, x, yq ´pρ λ pyqq ´1 i´1 ÿ j"0 ˆi j ˙pp λ q pjq pT, x, yq p λ pT, x, yq pρ λ q pi´jq pyq.

So an inductive argument using (6.18) and (1.4) will conclude (1.5) for all i ď k ´2.

Finally, consider Theorem 1.3 iv). By symmetry, the mapping x Þ Ñ pp λ q piq pT, x, yq is continuous for all T, y, locally uniformly in y. We conclude using (5.97) and (1.5) as in the proof of Theorem 5.1 iii).

In summary, to carry out the above inductive argument for Theorem 1.3, all we need to do is to verify the properties i)-vii) at each step. We first consider i), followed by iv) and then check ii), iii), v), vi) and vii). The ideas to show these properties at each step are similar. So we only check them for the j " 2 case in details and indicate the necessary modifications to make them work for the general case. 6.2. Proofs of the properties concerning φ j λ . Let λ P p´1, 1q Þ Ñ g λ P M k pM q be a C k curve pk ě 4q. Assume all the properties i)-vii) in Section 1.3 hold true for r φ i λ , φ i λ and pp λ q piq λ , i ă j ď k ´2. We continue to verify the conclusions for r φ j λ , φ j λ and pp λ q pjq λ .

Proof of properties i) and iv) in Section 6.1. We first show i) and the estimation in iv).

We begin with the case j " 2. For i), it suffices to consider the differentiability and L q integrability of each term in (5.96). We add an upper-script λ to pIq, pIIq, pIIIq and pIVq in (5.96) to indicate their dependence on λ.

For pIVq λ , it is differentiable in λ by Lemma 4.17 and Theorem 5.1 for the k " 3 case. Denote the differential by `pIVq λ ˘p1q λ . Then `pIVq λ ˘p1q λ " ´@Dπptu T s λ q p2q λ , ∇ λ ln p λ pT, x, tx T s λ q

D λ
´@Dπptu T s λ q p1q , ∇ λ pln p λ q p1q λ pT, x, tx T s λ q

D λ
´@Dπptu T s λ q p1q , ∇ λ ptx T s λ q p1q λ ∇ λ ln p λ pT, x, tx T s λ q D λ ": pIVq λ 1 `pIVq λ 2 `pIVq λ 3 .

By an abuse of notation, we use cpqq to denote a constant depending on T, T 0 , m, q, }g λ } C 4 and }X } C 3 , which may vary from line to line. Using i) of Proposition 4.27 and Lemma 4.13, we obtain cpqq such that

`Eˇˇp IVq λ 1 ˇˇq ˘2 ď E › › ptu T s λ q p2q λ › › 2q ¨E› › ∇ λ ln p λ pT, x, tx T s λ q › › 2q ď cpqq.
Similarly, by i) of Proposition 4.27 and Lemma 4.13, we can derive that `Eˇˇp IVq λ 3 ˇˇq ˘3 ď E › › ptu T s λ q p1q λ › › 4q ¨E› › ∇ λ,p2q ln p λ pT, x, tx T s λ q › › q ď cpqq. Using i) of Proposition 4.27 and (5.2), we obtain

`Eˇˇp

IVq λ 2 ˇˇq ˘2 ď E › › ptu T s λ q p1q λ › › 2q ¨E› › ∇ λ pln p λ q p1q λ pT, x, tx T s λ q › › 2q ď cpqq. As for the conditional expectations, by ii) of Proposition 4.27 and Lemma 4.13, we obtain Using this and ii) of Proposition 4.27, we conclude that the same type of bound is valid for the term p λ pT, x, yqE

P λ,x
,y,T ˇˇpIVq λ 2 ˇˇ.

By Lemma 4.17, λ Þ Ñ pIIIq λ is also differentiable in λ. Its differential is given by `pIIIq λ ˘p1q λ " @ Dπptu T s λ q ": pIIIq λ 1 `pIIIq λ 2 `pIIIq λ 3 , where the last term denotes the differential of the inner product. Then it is standard to estimate the expectation of `pIIIq λ ˘p1q λ using Hölder's inequality, Burkholder's inequality and i) of Proposition 4.27, which gives

´Eˇˇ`p

IIIq λ ˘p1q λ ˇˇq ¯2 ď cpT q `T 2q q ´E› › ptu T s λ q

p2q λ › › 2q `E› › ptu T s λ q p1q λ › › 2q `E› › ptu T s λ q p1q λ › › 4q ¯ď cpqq.
For the corresponding conditional expectation estimation, we use (4.39), Hölder's inequality and Burkholder's inequality as before. It is easy to deduce that For pIq λ , we can check the differentiability of ∇ λ T,V,s Dπptu T s λ q p1q λ term-by-term using its expression in Corollary 5.33. The estimation can be done as above using Proposition 4.27.

By the inductive construction, r φ j´1 λ pT, x, wq involves the mixed differentials of order j in λ and in ∇ λ T,s of tu T s λ and can be expressed by a multi-stochastic integral involving a mixture of differential processes t `tu t s λ ˘pj 1 q λ u j 1 ďj´1 , t " D p1q tF t,t s λ ptu t s λ , wq ‰ piq λ u iďj´2 and t∇ λ,plq pln p λ q piq λ pT, x, tx T s λ qu l`iďj´1,iďj´2 . So, by Lemma 4.17 and Proposition 4.27, we have the differentiability of λ Þ Ñ r φ j´1 λ pT, x, wq and the derivative p r φ j´1 λ q p1q λ pT, x, wq involves t `tu t s λ ˘pj 1 q λ u j 1 ďj , t " D p1q tF t,t s λ ptu t s λ , wq ‰ piq λ u iďj´1 and t∇ λ,plq pln p λ q piq λ pT, x, tx T s λ qu l`iďj,iďj´1 . The estimations in i) and iv) will follow from a repeated application of Proposition 4.27 to the multiple stochastic integral as in the j " 2 case. The bound estimation in iv) contains `T ´1d r g λ px, yq `p? T q ´1˘j since the formula of p r φ j´1 λ q p1q λ pT, x, wq contains the terms ∇ λ,pjq ln p λ pT, x, tx T s λ q, ∇ λ,pj´1q pln p λ q p1q λ pT, x, tx T s λ q. As to the continuity and its uniformity in T and λ of the map y Þ Ñ Ψ j pyq :" E ´p r φ j´1 λ q p1q λ pT, x, wq ˇˇtx T s λ pwq " y ¯,

we compare Ψ j pyq with Ψ j p r F pyqq, where t r F u rPR is the flow map generated by a bounded smooth vector field W on Ă M . Let t r Fs λ be as in Section 5 which extends r F to the r g λ -Brownian paths. Then, as in the proof of Theorem 5.1, we obtain Ψ j p r F pyqq " E ˜p r φ j´1 λ q p1q λ ˝tr Fs λ ¨dP λ x ˝tr Fs λ dP λ x ˇˇˇˇt x T s λ " y ¸¨p λ pT, x, yq p λ pT, x, r F pyqq ¨dVol λ dVol λ ˝rF pyq.

In the proof of Lemma 5.34, we obtained the local uniform boundedness in pT, yq and λ of for elements tu t s λ , t `tu t s λ ˘pj 1 q λ u j 1 ďj and t " D p1q tF t,t s λ ptu t s λ , wq ‰ piq λ u iďj´1 that appear in the expression of p r φ j´1 λ q p1q λ , which is true since they can be further reduced to the A appearing in Lemma 5.34 by the construction of t r Fs λ .

Proof of properties ii), iii) and v) in Section 6.1. Using (4.45) and the inductive assumption on the boundedness of E To show Φ j λ pY q is C 1 , we follow the argument in the proof of Lemma 5.34. Let tF s u sPR be the flow generated by a smooth bounded vector field V on Ă M . Let tF s s λ be constructed as in Section 5.5, which extends F s to Brownian paths starting from x up to time T using the auxiliary function s. Then the change of variable comparison in Section 5.1 gives Φ j λ pY qpF s yq " E The process Φ j λ pY, wq ˝tF s s λ is differentiable in s with pΦ j λ ˝tF s s λ q 1 s " @ ∇ Vptx s T s λ q Y ptx s T s λ q, r φ j´1 λ ˝tF s s λ Dπptu s T s λ q p1q λ D λ `@Y ptx s T s λ q, `∇s,λ T,V,s r φ j´1 λ ˘˝tF s s λ Dπptu T s λ q p1q λ D λ `@Y ptx s T s λ q, r φ j´1 λ ˝tF s s λ ∇ s,λ T,V,s Dπptu T s λ q p1q λ D λ and this differential is L q integrable conditioned on x T " y, uniformly in s, for all q ě 1. Using this and Proposition 5.20, we can conclude that Φ j λ pY qpF s yq is differentiable in s.

Following the proof of Lemma 5.34 (see (5.89)), we obtain pΦ j λ pY qpF s yqq 1 0 " E P λ x,y,T

´@∇ Vptx T s λ q Y, r φ j´1 λ ¨Dπptu T s λ q p1q λ pwq D λ `@Y ptx T s λ q, ∇ λ T,V,s `r φ j´1 λ ¨Dπptu T s λ q p1q λ pwq ˘Dλ `@Y ptx T s λ q, r φ j´1 λ ¨Dπptu T s λ q p1q λ pwq D λ Ð Ý E T,V,s (6.19)

": E To show y Þ Ñ pΦ j λ pY qpF s yqq 1 0 is continuous, we compare (6.19) with its value at nearby points. Choose another smooth bounded vector field W on Ă M and let t r F u rPR be the flow it generates and let t r Fs λ be its extension to r g λ -Brownian paths starting from x up to time T . A change of variable argument in Section 5.1 for t r Fs λ shows that for z " r F pyq, We can show the local uniform convergence (in py, T q and λ) of pΦ j λ pY qpF s zqq 1 0 to pΦ j λ pY qpF s yqq 1 0 as r Ñ 0 exactly as we did in the previous proofs of properties i) and iv).

As to the estimations in v), ˇˇz λ,j T pyq ˇˇcan be estimated using the conditional L q expectations of r φ j´1 λ pT, x, wq, ptu T s λ q p1q λ pwq, respectively. By (6.19), we have the formula in iii). By Claim 6.3, we obtain the formula of ∇ λ V z λ,j t pyq in (6.14). We can use them and Proposition 4.27 to give the desired estimation of Div λ z λ,j t pyq.

Proof of properties vi) and vii) in Section 6.1. The j " 1 case was considered in Theorem 5.1. When j " 2, since we have (6.1) for i " 1, 2, so, for all f P C 8 c p Ă M q, ż Ă M f pyqφ 2 λ pT, x, yqp λ pT, x, yq dVol λ pyq " ˆż Ă M f pyqφ 1 λ pT, x, yqp λ pT, x, yq dVol λ pyq ˙p1q λ .

This implies φ 2 λ pT, x, yq " pφ 1 λ q p1q λ pT, x, yq `φ1 λ pT, x, yq ¨´lnpp λ pT, x, yqρ λ pyqq ¯p1q λ " pφ 1 λ q p1q λ pT, x, yq ``φ 1 λ pT, x, yq ˘2 . Hence, pln p λ q p2q λ pT, x, yq " φ 2 λ pT, x, yq ´pφ 1 λ q 2 pT, x, yq ´pln ρ λ q p2q λ pyq and ∇pln p λ q p2q λ pT, x, yq " ∇φ 2 λ pT, x, yq ´2φ 1 λ pT, x, yq∇φ 1 λ pT, x, yq ´∇pln ρ λ q p2q λ pyq.

This, together with vi), vii) in the j " 1 case, shows that the estimation for the term |∇ plq pln p λ q p2q λ pT, x, yq| in (6.10) holds true if the same type of estimation is valid for |∇ plq φ 2 λ pT, x, yq|. By i)-v), Claims 6.2-6.3 apply. We have (6.20) φ 2 λ pT, x, yq " E ´r φ 2 λ pT, x, wq ˇˇtx T s λ pwq " y ¯, where r φ 2 λ pT, x, wq " `r φ 1 λ ˘p1q λ pT, x, wq´@∇ λ T,s r φ 1 λ pT, x, wq, Dπptu T s λ q p1q λ pwq D `r φ 1 λ pT, x, wq r φ 1 λ pT, x, wq. We can use (6.20) to derive the conditional expectation expressions of ∇ plq φ 2 λ pT, x, yq as in the proof of Theorem 5.1. Using this and Proposition 4.27, we can derive the desired estimations of ∇ plq φ 2 λ pT, x, yq and its L q -norm. For j ě 3, with i)-v), we have the identity ∇ plq pln p λ q pjq λ pT, x, yq " ∇ plq φ j λ pT, x, yq ´j´1 ÿ

i"1

∇ plq `φi Since f is arbitrary, we must have φ i λ pT, x, yq " pφ i´1 λ q p1q λ pT, x, yq `φi´1 λ pT, x, yq ¨´lnpp λ pT, x, yqρ λ pT, x, yqq ¯p1q λ " pφ i´1 λ q p1q λ pT, x, yq `φi´1 λ pT, x, yq ¨φ1 λ pT, x, yq. Using this relationship inductively, we obtain φ j λ pT, x, yq " pφ 1 λ pT, x, yqq A differentiation of this equation gives (6.21). By induction, we see that the differentials pφ i λ q prq λ pT, x, yq for r ď j´i´1 only consist of pln p λ q psq λ pT, x, yq, `ln ρ λ ˘psq λ pyq up to order s " i `r ď j ´1. By the inductive assumption on the gradient estimations of pln p λ q psq λ pT, x, yq, s ď j ´1, to obtain vii) for ˇˇ∇ plq pln p λ q pjq λ pT, x, yq ˇˇ, it suffices to give the estimation for |∇ plq φ j λ pT, x, yq|. By i)-v), Claims 6.2-6.3 apply and we have φ j λ pT, x, yq :" E ´r φ j λ pT, x, wq ˇˇtx T s λ pwq " y ¯,

where r φ j λ pT, x, wq is defined inductively using (6.8). As in the proof of Theorem 5.1, we can further obtain r φ j,plq λ pT, x, wq such that ∇ plq φ j λ pT, x, yq " E ´r φ j,plq λ pT, x, wq ˇˇtx T s λ pwq " y ¯.

The term r φ j,plq λ pT, x, wq involves the derivatives of r φ j λ pT, x, wq under r F up to the j-th order and can be formulated as a stochastic integral using s, s 1 , tus λ , t `tu t s λ ˘pj 1 q λ u j 1 ďj , t " D p1q tF t,t s λ ptu t s λ , wq ‰ piq λ u iďj´1 and t∇ λ,plq pln p λ q piq λ pT, x, tx T s λ qu l`iďj,iďj´1 . So we can use this and ii) of Proposition 4.27 to derive the desired bound of |∇ plq φ j λ pT, x, yq| as in Theorem 5.1. As to vi), it is equivalent to estimate `Ep} r φ j,plq λ pT, x, wq} q q ˘1 q , which can be handled using i) of Proposition 4.27 and the inductive assumption on (1.4) for i ă j.

Regularity of the entropy

The analog of formula (1.2) for the entropy involves the Martin kernel of the Brownian motion on p Ă M , r g λ q for g λ P k pM q. Recall that the Green function on p Ă M , r g λ q is given by G λ px, yq :" where c 0 can be chosen to be independent of λ for g λ in a small neighborhood of g 0 . By Anderson and Schoen [AS] (see also [Anc]), the Martin kernel for p Ă M , r g λ q is defined by (7.1) k λ px, y, ξq :" lim zÑξ k λ px, y, zq, where k λ px, y, zq :" G λ py, zq G λ px, zq .

Hence, the logarithm of the Martin kernel is an analog of the Busemann function using the Green metric since ln k λ px, y, ξq :" lim zÑξ `dG λ px, zq ´dG λ py, zq ˘, for x, y P Ă M , ξ P B Ă M .

Moreover, it is known that the entropy satisfies the following formula ([Kai1])

h λ " ´ż ∆ λ y ln k λ px, y, ξq ˇˇy "x dm λ px, ξq.

For x, y P Ă M fixed, the function k λ px, y, ξq is a continuous version of the Radon-Nikodyn derivative pd r m y {d r m x qpξq; the gradient Zpx, ξq :" ∇ λ y k λ px, y, ξq| y"x is a G-equivariant stable vector field that depends Hölder continuously on ξ with the Hölder exponent uniformly in λ for g λ in a small neighborhood of g 0 ( [AS], see also [Ha]). Furthermore, we have the following.

Lemma 7.1. Let M be a closed connected smooth manifold. For each g P k pM q (k ě 3), there exist a neighborhood V 2 g of g in k pM q and b 2 , b 2 ą 0, such that for any C k curve λ P p´1, 1q Þ Ñ g λ P V 2 g with g 0 " g, the second order differentials of k λ px, y, ξq in y at y " x are Hölder continuous in ξ with exponent b 2 ; for b ă b 2 κ, where κ is as in (3.1), we have (7.2) ∆ λ y ln k 0 px, y, ξq ˇˇy"x , p∆ λ y q 1 λ ln k 0 px, y, ξq ˇˇy"x P H 0 b .

Proof. The second part follows from [AS,Theorem 6.2] and the first part. We show the first part by following the proof of [START_REF] Hamenstädt | An explicit description of harmonic measure[END_REF]Lemma 3.2].

Let x P Ă M and let Bpx, δq be a small neighborhood around x with a positive radius δ. For v " px 1 , ζq P S Ă M r g with x 1 P Bpx, 2δqzBpx, δq and ρ, 0 ă ρ ď π{2, let C λ pv, ρq :"

! z P Ă M : = r g λ
x 1 pv, 9 γ r g λ

x 1 ,z p0qq ă ρ ) , Cpv, ρq :" C 0 pv, ρq be the open cone of vertex x 1 , axis v and angle ρ, where = r g λ

x 1 p¨, ¨q is the r g λ angle function in S x 1 Ă M and 9 γ r g λ

x 1 ,z p0q is the initial tangent vector of the r g λ unit speed geodesic γ r g λ

x 1 ,z from x 1 to z.

There exists a neighborhood V g of g in k pM q such that if g λ P V g , then for all v, Cpv, π{6q Ă C λ pv, π{4q Ă Cpv, π{3q and for all x P Ă M , Bpx, δ{4q Ă B g λ px, δ{2q Ă Bpx, δq. 4 Let tx s,t u |s|,|t|ă1 , with x 0,0 " x, be any C 2 two parameter family of points in Bpx, δ{4q. For Cpv, π{2q apart from Bpx, δq and z P Cpv, π{2q, let ϕ s,t pzq :" 1 st ´kλ px, x s,t , zq ´kλ px, x 0,t , zq ´kλ px, x s,0 , zq `kλ px, x 0,0 , zq ¯.

To conclude the first part of Lemma 7.1, it suffices to show for V g small, there is some C, C ą 0, independent of s, t, x, z and g λ such that (7.3) |ϕ s,t pzq| ď C.

4 There is a neighborhood Vg of g in R 3 pM q and a number r such that for r g, r g 1 P Vg, τ ě r, = r g 1

x 1 p 9 γ r g 1

x 1 ,γ r g 1 x,ξpτ q , 9 γ r g x 1 ,γ r g x,ξpτ q q ă = r g 1

x 1 p 9 γ r g 1

x 1 ,γ r g 1 x,ξprq , 9 γ r g x 1 ,γ r g x,ξprq q `π{100.

It suffices then to control the angles on Bpx, r `2δq.

This is because (7.3) implies that, for z P C λ pv, π{4q, ϕ s,t pzq `C " 1 st `Gλ px s,t , zq ´Gλ px 0,t , zq ´Gλ px s,0 , zq `Gλ px 0,0 , zq ˘`CG λ px, zq G λ px, zq is the quotient of two positive harmonic functions in C λ pv, π{4q which vanish at the infinity boundary B Ă M . Hence, by using [AS,Theorem 6.2], for V g small, we obtain two positive numbers C 1 , b 2 , independent of s, t, x and g λ , such that (7.4) ˇˇpϕ s,t pzq `Cq ´pϕ s,t pz 1 q `Cq ˇˇď C 1 e ´b2 κpz|z 1 q λ

x , @z, z 1 P C λ pv, π{4q.

Let ξ, η P B Ă M be points lying in the closure of Cpv, π{6q. Letting z Ñ ξ, z 1 Ñ η and then letting s, t Ñ 0 in (7.4), the first part statement of Lemma 7.1 follows by using (7.1).

It remains to show (7.3), or, equivalently, (7.5) ˇˇˇ1 st ´Gλ px s,t , zq ´Gλ px 0,t , zq ´Gλ px s,0 , zq `Gλ px 0,0 , zq ¯ˇˇˇď CG λ px, zq.

Since G λ p¨, zq is harmonic in B g λ px, δ{2q, by the Harnack inequality ( [AS]) and the infinitesimal Harnack inequality of Cheng-Yau ([CY]), for V g small, there is some C 2 , C 2 ą 0, independent of s, t, x, z and g λ such that G λ py, zq, › › ∇ λ y G λ py, zq › › r g λ ď C 2 G λ px, zq, @y P B g λ px, δ{2q, z P C λ pv, π{4q. To continue, we consider L W ˇˇy G λ py, zq, where W is any smooth bounded vector field on Ă M , L W ˇˇy is the Lie derivative in W evaluated at y. Then, in the distribution sense, ∆ λ L W ˇˇy G λ py, zq " L W ˇˇy ∆ λ G λ py, zq `"L W , ∆ λ ‰ G λ py, zq " " L W , ∆ λ ‰ G λ py, zq,

where the last commutator term is a linear combination of the contractions R λ ˚∇λ G λ p¨, zq, ∇ λ R λ ˚Gλ p¨, zq evaluated at W P T y Ă M . Since L W ˇˇy G λ py, zq is C 1 in y, it must be a real solution function f to the equation ∆ λ f pyq " " L W , ∆ λ ‰ G λ py, zq.

Hence the classical estimation property of elliptic equation (cf. [Fr]) shows that there is some positive C 3 depending on the geometry, which can be chosen to be independent of x, z, g λ for V g small, such that This shows (7.5) since W can be arbitrary.

Proof of Theorem 1.6. Let V 2 g , b 2 be as in Lemma 7.1. Let b ă b 2 κ, V g and H 0 b be such that Theorem 1.2 holds true. Let λ P p´1, 1q Þ Ñ g λ P V 2 g X V g with g 0 " g be a C 3 curve. We omit the index 0 for h 0 , k 0 , p 0 , Z 0 , ∆ 0 , Div 0 , ∇ 0 , x¨, ¨y0 , Vol 0 and m 0 at g 0 .

We study the differentiability of h λ by writing, as in [START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF], 1 λ ph λ ´hq " 1 λ ph λ ´hλ,0 q `1 λ ph λ,0 ´hq ": pIq λ `pIIq λ , where h λ,0 " ´ż ∆ λ y ln kpx, y, ξq ˇˇy "x dm λ px, ξq.

Then, by (7.2) and Theorem 1.2, lim λÑ0 pIIq λ " ´ż p∆ λ y q 1 0 ln kpx, y, ξq ˇˇy"x dmpx, ξq ´ż ∆ y ln kpx, y, ξq ˇˇy"x dpm λ q 1 0 px, ξq.

Using u 1 for the function such that ∆u 1 " ´∆y ln kpx, y, ξq ˇˇy"x ´h, psee [LS2, (5.7)]q, we obtain, as in Section 3.3, K :" lim λÑ0 pIIq λ " ż ˆ´1 2 x∇traceX , Z `∇u 1 y `Div `X pZ `∇u 1 q ˘˙dm.

Clearly, K is linear on X P C k pS 2 T ˚q. When g is locally symmetric, u 1 " 0 and Z " X. Hence, K " ż ˆ´1 2 x∇traceX , Xy ´ X pX, Xq ˙dm, which vanishes (see Remark 3.12).

We will now show lim λÑ0 pIq λ " 0, which will complete the proof of Theorem 1.6. Following [START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF]Proposition 2.4], we obtain h λ,0 " inf są0 th λ,0 psqu, where h λ,0 psq :" lim tÑ8 ´1 t ż pln ppst, x, yqqp λ pt, x, yq dVol λ pyq.

Then, for all λ ą 0, 1 n ż ln ppn, x, yq p λ pn, x, yqρ λ pyq p λ pn, x, yq dVol λ pyq.

pIq
To estimate pIq λ , consider the stationary Markov chain on the space Ω " Ă M NYt0u with transition probability p λ p1, x, yq dVol λ pyq and the process Y 0 pwq " 1 and, for n ě 1, Y n pwq :" pp1, w 0 , w 1 q p λ p1, w 0 , w 1 qρ λ pw 1 q ¨pp1, w 1 , w 2 q p λ p1, w 1 , w 2 qρ λ pw 2 q ¨¨¨p p1, w n´1 , w n q p λ p1, w n´1 , w n qρ λ pw n q .

Observe that ppn, x, yq p λ pn, x, yqρ λ pyq " E P λ x `Yn pwq ˇˇw n " y ˘.

So we may write Set w i " y and let pty t s λ , t t s λ q tě0 be the stochastic pair in Ă M ˆOr g λ p Ă M q that defines the r g λ Brownian motion on Ă M starting from y. Then,

pIq
E P λ w i ˆln Y i`1 pwq Y i pwq
˙" E Q ˆln pp1, y, ty 1 s λ pwqq p λ p1, y, ty 1 s λ pwqqρ λ pty 1 s λ pwqq ˙": E Q ppIIIq λ,y q , which is L 1 integrable in y. Hence the ergodic theorem applied to the g λ Brownian motion on M (see Proposition 2.2) gives that lim nPN,nÑ8

1 n n´1 ÿ i"0

E P λ x ˆEP λ w i ˆln Y i`1 pwq Y i pwq ˙˙" E P λ E Q `pIIIq λ,w 0 ˘.
Since E Q ppIIIq λ,y q 1 λ " E Q ´´pln p λ q 1 λ p1, y, ty 1 s λ pwqq ´pln ρ λ q 1 λ pty 1 s λ pwqq ĒQ ˜@ ∇ z ln pp1, y, zq p λ p1, y, zqρ λ pzq ˇˇˇz "ty 1 s λ pwq , Dπ `t 1 s λ ˘p1q λ pwq D ¸,

we conclude that ppIIIq λ,y q 1 λ is L 1 integrable, uniformly in λ and y, by using Theorem 1.3 ii), Lemma 4.13 and Proposition 4.27 i) for pty 1 s λ , t 1 s λ q. Moreover, pE Q ppIIIq λ,y qq 1 0 " E Q ´´pln p λ q 1 0 p1, y, ty 1 s 0 q ´pln ρ λ q 1 0 pty 1 s 0 q ¯" 0 by taking the differential in λ of ş p λ p1, y, zq dVol λ pzq " 1.

Hence, lim λÑ0`0

1 λ E P λ E Q `pIIIq λ,w 0 ˘" E P `EQ `pIIIq λ,w 0 ˘˘1 0 " 0, where the first equality holds since we are integrating a function that depends only on w 0 . Consequently, we obtain lim λÑ0`0 pIq λ " 0. In the same way, we show lim λÑ0´0 pIq λ " 0. Thus, lim λÑ0 pIq λ " 0.

Remark 7.2. Note that for all λ, h λ ď pυ λ q 2 by [Gu] and [START_REF] Kaimanovich | Brownian motion and harmonic functions on covering manifolds. An entropic approach[END_REF]. As in Corollary 3.10, we can also use [BCG], [KKPW] and the C 1 differentiability of λ Þ Ñ h λ for any C 3 curve λ Þ Ñ g λ P 3 pM q to conclude that ph λ q 1 0 " 0 at locally symmetric g 0 .

In proving Theorem 1.6, we obtain the following formula.

Theorem 7.3. Let M be a closed connected smooth manifold and let g P 3 pM q. For any C 3 curve λ P p´1, 1q Þ Ñ g λ P 3 pM q with g 0 " g and constant volume, ph λ q 1 0 " ż ˆ´1 2 x∇traceX , Z `∇u 1 y `Div `X pZ `∇u 1 q ˘˙dm.

  In this paper, we are mainly interested in the behaviors of the following two dynamical quantities. One is the linear drift

  pγq, h µ :" lim nÑ`8 ´1 n ÿ γPG µ pnq pγq log µ pnq pγq,

r g 0

 0 ˆdp´1,1q , where d r g 0 is the induced metric of d r g 0 in Fp Ă

Lemma 5. 5 .

 5 Assume there are locally L 1 integrable functions tφ 1 λ pT, x, yqu xP Ă M ,T PR `on Ă M which are continuous in λ-parameter and are continuous in pT, yq-parameter, locally uniformly in λ, such that, for any f P C 8

λ

  {dVol 0 and the second equality holds by Fubini theorem. Note that if a continuous function φ is such that ş Ă M φpyqf pyq dVolpyq " 0 for all f P C 8 c p Ă M q and a volume element Vol of a C 2 Riemannian metric, then φ is zero. Hence we can conclude from (5.5) that (5.6) p λ pT, x, yqρ λ pyq ´p0 pT, x, yqρ 0 pyq "

λ 1 λ

 1 continuous with respect to P λ x . Clearly, for any bounded measurable function f on Ă M , E ´Φ1 λ pY, wqf ptx T s λ pwqq ¯"E ´Φ1 λ pY qpyqf pyq (5.17) pY qpF s pyqqf pF s pyqqp λ pT, x, F s yq dVol λ pF s pyqq,(5.18) where the first equality holds by the definition of conditional measures and the second equality holds by changing the variable to F s pyq. The left hand side of (5.17), after a change of variable under F s , is equal to λ pwq " y ¸¨f pF s pyqqp λ pT, x, yq dVol λ pyq.(5.19) Since f is arbitrary, a comparison of (5.18) with (5.19) implies that Φ pY qpF s pyqq "E ´Φ1 λ ˇˇtx T s λ pwq " F s pyq λ pwq " y ¸pλ pT, x, yq p λ pT, x, F s yq dVol λ dVol λ ˝F s pyq. (5.20) Note that p λ pT, x, yq and the volume element Vol λ are C k in the y variable. So the differentiability in the s parameter of Φ 1 λ pY qpF s pyqq will follow from the differentiability in the s

  suffices to show for any flow F s generated by a smooth bounded vector field V on Ă M , s Þ Ñ Φ 1 λ pY qpF s yq, y P Ă M , is differentiable at s " 0 and the differential y. Let tF s s λ be introduced in Section 5.5, which extends F s to Brownian paths starting from x up to time T using the auxiliary function s. By Proposition 5.23, P λ x ˝tF s s λ is absolutely continuous with respect to P λ x . So the change of variable comparison in Section 5.1 works, which gives (5.20), i.e., wq ˝tF s s λ ¨dP λ x ˝tF s s λ dP λ x ¸pλ pT, x, yq p λ pt, x, F s yq dVol λ dVol λ ˝F s pyq,

  ,T , where s is given in Section 5.2. So, by (5.20) and (5.88), we have

  Since p λ and Vol λ are continuous in y, for continuity of pΦ 1 λ pY qpF s yqq 1 0 in y, it remains to show the conditional expectation of the following difference tends to 0 as r goes to 0:Ψ 1 λ pY, Vq ˝tr Fs λ ¨dP | r pIq 2 | 2 islocally uniformly bounded in r by Proposition 5.23 and E P λ x,y,T |pIIq 2 | 2 is bounded by using Proposition 5.20 and Proposition 5.29. Note that |Y |, |∇ V Y | are locally bounded at y and the difference between Y pzq and Y pyq, ∇ V Y pzq and ∇ V Y pyq under parallel transportation along pı Þ Ñ ı F pyqq ıPr0,rs is bounded by a multiple of r. Using this, (5.89) and a standard split argument by Hölder's inequality, we see that to conclude the first property in (5.90), it suffices to show r pIIIq :" Dπptu T s λ q p1q λ

  1 p r pVIq `rpVIIqq .Clearly, r pVIIq Ñ 0 as r Ñ 0. For the second property in (5.90), it remains to show r pVIq Ñ 0 as r Ñ 0. Following the proof of Proposition 5.20, we obtain r g τ s λ pwq, dBτ pwqy`1 4 ş T 0 |t r g τ s λ pwq| 2 dτ u ": r E T pwq and p r E T pwqq 1 r " r E T pwq ¨ˆ´1 2 ż T 0

6. 1 .

 1 A sketch of the proof for Theorem 1.3 with i ě 2. Lemma 6.1. The i) of Theorem 1.3 holds true if there are locally absolutely integrable functions tφ i λ pT, x, yqu xP Ă M ,T PR `,iďk´2 on Ă M , which are continuous in the λ-parameter and are continuous in the pT, yq-parameter, locally uniformly in λ, such that for any f P C 8 c p Ă M q,

  › ∇ λ,p2q ln p λ pT, ,yqq . By Corollary 5.36, for some different c, c, p λ pT, x, yq › › ∇ λ pln p λ q

  T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d `tu T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ T s λ ptu τ s λ q ´1Ric ´1 tuτ s λ d Ð Ý B τ D p1q λ

  ln ppT ´τ, tx τ s λ , yq

  uniformity in pT, yq and λ of the convergence of as r Ñ 0.Following the estimation for iv), we obtain the local uniform boundedness in pT, yq and λ of x, wq ˝tr Fs λ › › 2 .So for iv), it remains to show the local uniform convergence in pT, yq and λ of using ii) of Proposition 4.27, can be reduced to showing the local uniformity in pT, yq and λ of the convergence of

  :" E ´@Y ptx T s λ pwqq, r φ j´1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq D λ ˇˇtx T s λ pwq " y īs a locally bounded functional on C k bounded vector fields Y on Ă M .

  wq ˝tF s s λ ¨dP λ x ˝tF s s λ dP λ x ¸pλ pT, x, yq p λ pt, x, F s yq dVol λ dVol λ ˝F s pyq, where Φ j λ pY, wq " @ Y ptx T s λ pwqq, r φ j´1 λ pT, x, wq ¨Dπptu T s λ q p1q λ pwq D λ .

  Vq ˝tr Fs λ ¨dP λ x ˝tr Fs λ dP λ x ¸pλ pT, x, yq p λ pT, x, zq dVol λ dVol λ ˝rF pyq.

  x, yq ´∇plq `ln ρ λ ˘pjq λ pyq. (6.21) By Theorem 5.1 i),φ 1 λ pT, x, yq " ´lnpp λ pT, x, yqρ λ pT, x, yqq ¯p1q λ " pln p λ q p1q λ pT, x, yq `pln ρ λ q p1q λ pyq.By (6.1), for all i, i ď j, and allf P C 8 c p Ă M q, ż Ă Mf pyqφ i λ pT, x, yqp λ pT, x, yq dVol λ pyq " ˆż Ă M f pyqφ i´1 λ pT, x, yqp λ pT, x, yq dVol λ pyq ˙p1q λ .

  , yq ´`ln ρ λ ˘pjq λ pyq.

ż 8 0 p

 0 λ pt, x, yq dt, for x, y P Ă M , and it can be associated with a "Green metric" on Ă M ([START_REF] Ledrappier | Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes[END_REF]) by lettingd G λ px, yq :" " ´lnpc 0 G λ px, yqq, if d r g λ px, yq ą 1, ´ln c 0 , otherwise,

G

  λ py, zq ď2C 3 C 2 G λ px, zq.

  W s px, ξq :" Each W s px, ξq coincides with the collection of the initial speed vectors of the geodesics asymptotic to ξ and can be identified with Ă M . Associated with E ss are the strong stable manifolds

	" py, ηq P Ă M ˆB Ă M : lim sup tÑ`8 ln dist pΦ (2.2) 1 t W ss px, ξq :" " py, ηq P Ă M ˆB Ă M : lim sup tÑ`8 1 t ln dist pΦ

t py, ηq, Φ t px, ξqq ď 0 * . t py, ηq, Φ t px, ξqq ă 0 * . Each W ss px, ξq, locally, is a C k´1 graph from E ss px,ξq to E c px,ξq ' E su px,ξq and is tangent to E ss px,ξq ([SFL]). It is true that Φ t pW ss px, ξqq " W ss pΦ t px, ξqq and the union of these images is just the stable manifold, i.e., W s px, ξq " ď tPR Φ t pW ss px, ξqq . The weak and strong unstable manifolds, denoted by W u px, ξq and W su px, ξq, respectively, can be defined similarly as in (2.1) and (2.2) by reversing the time. They have tangents E cu :" E c ' E su and E su , respectively.

  Then there exist λ n P r´1, 1s, T n P R `, T n Ñ 8, and points x n , ξ n , η n , ξ n " η n , such that By definition of the Gromov product p¨|¨q, for all ξ " η P B Ă M , y, z P M 0 and λ P r´1, 1s, ˇˇpξ|ηq y ´pξ|ηq z ˇˇď 2dpy, zq ď Const. ¨dλ py, zq,

	(3.3)	1 T n	E λn xn,ξn ´pξ n ˇˇη n q x λn Tn	´pξ n ˇˇη n q xn	¯ă 1 4	.
	where the constant is independent of λ, ξ, η, y and z. Hence by uniform continuity of
	λ Þ Ñ p λ pt, x, ¨q in x, we can find t 0 small enough such that
	(3.4)	sup λPr´1,1s	sup 0ďtďt 0	sup x,ξ	sup η "ξ	E λ x,ξ

  psq ds, i " 1, ¨¨¨, m.Let w ptq " pw 1 ptq, ¨¨¨, w m ptqq. We see that t Þ Ñ w ptq is smooth and satisfies

	lim Ñ0	sup tPR `› › w ptq ´wptq	› › " 0.
	As w varies, B t : w Þ Ñ w ptq defines an F t` -measurable process on Θ `. Each B t solves
	d dt	p r B t q "	m ÿ i"1	e i p r B t q	¨d dt	pw i ptqq
	and, almost surely, the limit of B t (as Ñ 0) gives the Brownian motion B t ([

  pwq| 2q d|z t pwq| 2q ´1 2|z t pwq| 4q xd|z t pwq| 2q , d|z t pwq| 2q y. Note that tM j u, N all have norms bounded by some constant depending on R, ∇R. Hence,

27) dz t pwq " m ÿ j"1 ´Mj pu t qz t pwq dB j t pwq `rM j pu t qs 2 zt pwq dt ¯`Npu t qz t pwq dt. (The coefficient of N in (4.27) is different from that in [Mal2, Theorem 5.1] since we are considering Brownian motion with generator ∆ instead of ∆{2.) By Itô's formula, d|z t pwq| 2q "2q|z t pwq| 2pq´1q xz t pwq, dz t pwqy `q|z t pwq| 2pq´1q xdz t pwq, dz t pwqy `2qpq ´1q|z t pwq| 2pq´2q xz t pwq, dz t pwqy 2 ; d ln |z t pwq| 2q " 1 |z t (4.28) |z t pwq| 2q " e ş t 0 d ln |ztpwq| 2q

  T . By the Markov property of p,

	(4.36)	dP x dP x,y,T	ˇˇˇF
	Hence
		dP x,y,T	ˇˇˇF
		dP x

t " ppT ´t, x t , yq ppT, x, yq " r ppT ´t, u t , yq ppT, x, yq ": Ξ t , @t P r0, T q, where r ppt, u, yq :" ppt, πpuq, yq. Using (4.7) and the heat equation, one can calculate using Itô's formula to obtain d ln Ξ t " xu ´1 t ∇ H ln r ppT ´t, u t , yq, dB t y ´}∇ H ln r ppT ´t, u t , yq} 2 dt. t " e t ş t 0 xu ´1 t ∇ H ln r ppT ´τ,uτ ,yq,dBτ y´ş t 0 }∇ H ln r ppT ´τ,uτ ,yq} 2 dτ u . Comparing this with (4.35), it implies b t :" B t ´2 ż t 0 u ´1 τ ∇ H ln r ppT ´τ, u τ , yq dτ is a Brownian motion with respect to P x,y,T and hence Proposition 4.15 holds for t P r0, T q.

  The maps tDF s t 1 ,t 2 u 0ďt 1 ăt 2 ďT are said to be C 1 in s if, for almost all w and any pυ s t 1

	and its Itô form is
	$ ' ' &	dθpv α t q " `vα t ˘dα t `Ric d pv α t q " pβ α t q ´1R ´βα t dα t , β `βα t θpv α t q ˘dt, α t θpv α t q ¯βα t `pβ	α t q ´1R	´βα t e i , pv α t qe i	¯βα t dt
	' ' %		`pβ	α t q ´1p∇pβ	α t e i qRq	´βα t e i , β	α t θpv α t q	¯βα t dt.
	Let α s P A be a one parameter family of random processes. We abbreviate
			β	s t :" β	α s t , F s t 1 ,t 2 :" F α s t 1 ,t 2 and DF s t 1 ,t 2 :" DF α s t 1 ,t 2 .
								with pβ	α t q tPrt 1 ,t 2 s solving the Stratonovich SDE
	(5.40)						dβ	α t " Hpβ	α t , ˝dα t q.
	By Proposition 4.3, F α t 1 ,t 2 are C k´2 diffeomorphisms for almost all w and the first order tangent map DF α t 1 ,t 2 satisfies the following (see also Lemma 4.4).
								Fp Ă M q, v α t :"
	"	DF α t 1 ,t	`βα t 1 , w ˘‰v α t 1 , t P rt 1 , T s satisfies the Stratonovich SDE
								dv α t " `∇pv α t qH ˘pβ	α t , ˝dα t q.
	In the pθ, q-coordinate, we have
					#	d d	`θ`v α t ˘˘" `vα t ˘˝dα t , ` `vα t ˘˘" pβ α t q ´1R ´βα t ˝dα t , β	α t	`θ`v α t	˘˘¯β α t

Lemma 5.13. Let α P A. For almost all w, any t 1 P r0, T s and v α t 1 P T β α t 1

  For }g s,n ´gs,n´1 }, we can use the inductive definitions of g s,n and g s,n´1 to compute that

	(5.47)						´α,n´2 } d.
	g s,n t	´gs,n´1 t	" ´Os,n t	´Os,n´1 t	¯ż s	rO ,n t s ´1R V,α ,n´1 ptq d
							0
			`Os,n´1 t	ż s	rO ,n t s ´1 ´O,n´1 t	´O,n t ¯rO ,n´1 t	s ´1R V,α ,n´1 ptq d
					0
			`Os,n´1 t	ż s	rO ,n´1 t	s
					0

´1

`RV,α ,n´1 ptq ´RV,α ,n´2 ptq ˘d ": paq t `pbq t `pcq t .

  dB τ has the same Brownian distribution as B t since O s are orthogonal frames and the distribution of a Euclidean Brownian motion is invariant under orthogonal transfers. So α s only differs from a Brownian motion by a drift term Since |g s | is bounded from above by a multiple of s ¨sup |V|, the Novikov's condition is satisfied. Hence the classical Carmeron-Martin-Girsanov Theorem says that the distribution of α s under r Q s is the same as Q, i.e., for any measurable subset A of C 0 pr0, T s, R

		s t pwq "	ż t	O s τ pwq dB τ pwq	`ż t	g s τ pwq dτ.
					0	0
	The process	ş t 0 O s τ ş t 0 g s τ dτ . Let
		M s t pwq :" e t´1 2	ş t 0 xg s τ pwq,O s τ pwqdBτ pwqy´1 4	ş t 0 |g s τ pwq| 2 dτ u
	and consider a new distribution r Q s on C 0 pr0, T s, R m q which is given by
						d r Q s dQ	pwq " M s T pwq.
	(5.65)			dQ s dQ	pwq "	1 T pα ´spwqq M s	.
	Note that the process M s t satisfies the equation
		dM s t "	´1 2	M s t xg s t pwq, O s t pwqdB t pwqy .
	So, by Itô's formula,
	(5.66) ´d ln M s t pα ´spwqq "	1 2	xg s t pα ´spwqq, O s t pα ´spwqq dα ´s t pwqy	`1 4	|g s t pα ´spwqq| 2 dt,
	where the second term of the right hand side of (5.66) has coefficient 1{4 since α ´s t	has
	variance 2t. On the other hand, we have
	(5.67)	α

m q, Q ptw P Auq " r Q s ptα s pwq P Auq , which, by a change of variable, gives

Q ptw P Auq " Q s `tM s pα ´spwqq : w P Au ˘.

Since A is arbitrary, this means Q and Q s are equivalent and s ˝α´s pwq " w " Bpwq, for almost all w.

  s t pα ´spwqq dα ´spwq `gs t pα ´spwqq dt " dB t pwq.

	So, (5.66) is also of the form
		´d ln M s t pα ´spwqq "	1 2	xg s t pα ´spwqq, dB t pwqy	´1 4	|g s t pα ´spwqq| 2 dt
	and hence	1 T pα ´spwqq M s	" e t 1 2	ş T 0 xg s t pα ´spwqq, dBtpwqy´1 4	ş T 0 |g s

t pα ´spwqq| 2 dtu .

  As a corollary of Proposition 5.19, we have P F s y ˝Fs y is equivalent to P y with dP F s y ˝Fs

									`Fs
									s
									0
	dP y	y	pβq "	dP F s y dP y ˝pF s y q ´1 ˝Fs y pβq "	dP F s y dP s	˝Fs y pβq "	dQ 0 dQ s	`αs pwq	˘.
			dP y	y	pβq " e t´1 2	ş T 0 xg s τ pwq, dBτ pwqy`1 4	ş T 0 |g s τ pwq| 2 dτ u ,
	where
	g s t pwq :"

y pβq ˘" α s pwq. Note that dα s pwq " O s pwqdBpwq `gs pwqdt. So, by (5.63) and (5.35), we have dP F s y ˝Fs

  For t, t, 0 ď t ă t ď T , we abbreviate

	rDF s t,t p s t , wqs :" rDF α s t,t p s t , wqs,
	r Č pDF s t,t qp s 0 , wqs :" pθ, ωq s t rDF s t,t p s t , wqspθ, ωq ´1 s t

  22, for β P C x,y pr0, T s, Ă

	where
				E	s T "	´1 2	ż T 0	@	pg s t q 1 s pwq, dB t pwq D	`1 2	ż T 0	@	pg s t q 1 s pwq, g s t pwq	D	dt,
	and both `dP	λ F s y,x,T ˝tF s s λ {dP	λ y,x,T ˘and E	s
													M q,
	dP λ x ˝tF s s λ dP λ x	pβq "	dP	λ F s y,x,T ˝tF s s λ dP λ y,x,T	pβq	¨pλ pT, x, F s yq p λ pT, x, yq	¨dVol λ ˝F s dVol λ pyq.
	By Proposition 5.20, dP	λ F s y,x,T ˝tF s s λ {dP	λ y,x,T is differentiable in s with
	´dP	λ F s y,x,T ˝tF s s λ {dP	λ y,x,T ¯1 s	" ´dP	λ F s y,x,T ˝tF s s λ {dP	λ y,x,T ¯¨E	s T ,

  So differentiating both sides in r at r " 0 gives∇ λ W pyq φ 1 λ pT, x, ¨q " φ 1 λ pT, x, yq´`ln p λ pT, x, r F pyqq ˘1 0 ``ln ρ λ p r F pyqq ˘1 0 T " y are L q pq ě 1q integrable, locally uniformly in the r parameter. Using Hölder's inequality, if we can further show

								Ē˜r
						φ 1 λ	˝tr Fs λ ¨dP λ x	˝tr Fs λ x dP λ	ˇˇˇˇt x T s λ " y ¸¸1 0	.
	It was shown in Proposition 5.23 that dP	λ x	˝tr Fs λ {dP	λ x is differentiable in r with
		`dP	λ x	˝tr Fs λ {dP λ x ˘1 r " r E t ¨`dP	λ x	x ˝tr Fs λ {dP λ	ȃnd
	both r E t and `dP x ˘conditioned on x ‹) r λ x ˝tr Fs λ {dP λ φ 1 λ ˝tr Fs λ is also differentiable in r with both r φ 1 λ ˝tr Fs λ and p r φ 1 λ
	1 λ	˝tr Fs λ ¨dP	λ x	˝tr Fs λ dP λ x	ˇˇˇˇt x T s λ " y ¸¨p λ pT, x, yq p λ pT, x, r F pyqq	¨dVol λ dVol λ ˝rF	pyq.

  ˆ1 ´ż ppst, x, yq dVolpyq ˙ď 0, where the third inequality holds since ´ln a ď a ´1 ´1 for all a ą 0. On the other hand,

	pIq λ ě	1 λ	´hλ ´hλ,0 p1q "
		1 λ	lim tÑ8 ´1 t	ż	ln	p λ pt, x, yq ppt, x, yq	p λ pt, x, yq dVol λ pyq
	"	1 λ	lim nPN,nÑ8
			λ "	1 λ	sup są0	lim tÑ8 ´1 t	ż	ln	p λ pt, x, yq ppst, x, yq	p λ pt, x, yq dVol λ pyq
			"	1 λ	sup są0	lim tÑ8 ´1 t	ż	ln	p λ pt, x, yqρ λ pyq ppst, x, yq	p λ pt, x, yq dVol λ pyq
			ď	1 λ	sup są0	lim tÑ8	1 t

The upper script 0,2 is to indicate that Φ 0,2 θ , Φ 0,2 are associated with `rD p0q tFtp¨, wqs

We use the upper script 1,1 to indicate the functions Φ 1,1 θ , Φ 1,1 are associated with `rD p1q tFtp¨, wqs

Acknowledgments -We thank MSRI, ICERM and IML for their partial support. The second author would like to also thank LPMA and the Department of Mathematics of the University of Notre Dame for hospitality during her stays.

The second author was partially supported by NSFC (No.11331007 and No.11422104) and Beijing Higher Education Young Elite Teacher Project (YETP0003).