Marc Arnaudon

Koléhè Coulibaly-Pasquier

Kol Éh

È Coulibaly-Pasquier

Laurent Miclo
email: laurent.miclo@math.cnrs.fr

COUPLINGS OF BROWNIAN MOTIONS WITH SET-VALUED DUAL PROCESSES ON RIEMANNIAN MANIFOLDS

Keywords: . Brownian motions on Riemannian manifolds, intertwining relations, setvalued dual processes, couplings of primal and dual processes, stochastic mean curvature evolutions, boundary and skeleton local times, generalized Pitman theorem. MSC2020 primary: 60J60, secondary: 60J65, 60H10, 58J65, 53E10, 60J55, 35K93

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Couplings of brownian motions with set-valued dual processes on riemannian manifolds

INTRODUCTION AND MAIN RESULTS

Markov intertwinings were introduced by Rogers and Pitman [START_REF] Rogers | Markov functions[END_REF] to give a direct proof of the famous relation between the Brownian motion and the Bessel-3 process due to Pitman [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF]. These relations were next used by Yor and his coauthors (see e.g. [START_REF] Yor | Intertwinings of Bessel processes[END_REF][START_REF] Carmona | Beta-gamma random variables and intertwining relations between certain Markov processes[END_REF]) to get identities in law and by Diaconis and Fill [START_REF]Strong stationary times via a new form of duality[END_REF] to construct strong stationary times. For a historical account of the subsequent development of the Markov intertwining technique, consult for instance Pal and Shkolnikov [START_REF] Pal | Intertwining diffusions and wave equations[END_REF].

At an algebraic level, a Markov intertwining relation is a (directed) weak similar relation, from a Markov semi-group p Pt q tě0 on a measurable state space p M , Mq to another Markov semi-group pP t q tě0 on a measurable state space pM, Mq, consisting of a Markov kernel (called the link) Λ from p M , Mq to pM, Mq such that @ t ě 0, Pt Λ " ΛP t (1.1) in the sense of the composition of Markov kernels. Depending on non-degeneracy properties of Λ, such a relation is more or less strong. Especially when Markov semi-groups are described by their generators, (1.1) is often replaced by

LΛ " ΛL (1.2)
where L and L are respectively the generators of p Pt q tě0 and pP t q tě0 . But then one has to be more careful with the meaning of generators (e.g. in the sense of martingale problems) and their domains, in particular the domains are transported via (1.2).

To be more useful from a probabilist point of view, it is convenient to convert (1.2) into a coupling between X :" p Xt q tě0 and X :" pX t q tě0 , two Markov processes respectively associated to L and L (called the dual and primal processes), so that the following relations hold for the conditional laws: @ t ě 0, LpX t | Xr0,ts q " Λp Xt , ¨q (1.3) In addition, one asks that p Xt q tě0 can be constructed from pX t q tě0 in an adapted way, meaning @ t ě 0, Lp Xr0,ts |Xq " Lp Xr0,ts |X r0,ts q (1.4)

Yor was wondering about such couplings between some piecewise linear Markov processes and squared Bessel processes, in order to simplify his approach to certain properties of the former processes similar to those of the latter, see the end of the introduction of [START_REF] Yor | Intertwinings of Bessel processes[END_REF].

Such couplings are crucial for the constructions of strong stationary times, as explained by Diaconis and Fill [START_REF]Strong stationary times via a new form of duality[END_REF] in a discrete time and finite setting. More precisely, in this situation X is an ergodic Markov chain with invariant probability π and X is a Markov chain absorbed in a unique point. A strong stationary time τ for X is a finite stopping time for X (and some independent randomness) such that τ and X τ are independent and X τ is distributed according to π. Taking into account (1.3) and (1.4), one can see that the absorption time for X is a strong stationary time for X.

Strong stationary times are important for two reasons (cf. Diaconis and Fill [START_REF]Strong stationary times via a new form of duality[END_REF]):

-They enable to sample exactly the invariant probability π, contrary to the usual approximations provided by Monte Carlo techniques.

-They provide a probabilistic alternative to functional analysis approaches for the quantitative investigation of convergence to equilibrium. More precisely, for any strong stationary time τ , we have @ t ě 0, spLpX t q, πq ď Prτ ą ts where the separation discrepancy spµ, πq between two probability measures µ and π is defined by spµ, πq :" ess sup π ˆ1 ´dµ dπ (where dµ{dπ is the Radon-Nikodym density). The separation discrepancy dominates the total variation norm and gives positivity properties of µ with respect to π. In the context of convergence to equilibrium, it is very difficult to estimate the discrepancy of spLpX t q, πq via functional inequality techniques (see e.g. the book [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] of Bakry, Gentil and Ledoux).

In the objective of constructing strong stationary times via intertwining duality, there are particular dual processes X which are taking values in V, the set of measurable subsets of M , but in general V is only a subset of V, consisting in some regular subsets. The absorption set is the whole set M . The heuristic goal of intertwining duality is then to construct random subsets Xt Ă V such that X t is already at equilibrium in Xt , for all t ě 0, in such a way that X is itself Markovian and ends up covering the whole state space M .

In the diffusion context, set-valued intertwining dual processes started to be constructed in Fill and Lyzinski [START_REF] Allen | Strong stationary duality for diffusion processes[END_REF] and [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF]. In [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF], set-valued dual processes for diffusions on Riemannian manifolds were identified as stochastic perturbations of mean-curvature flows. But the coupling of primal and dual processes were not considered in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] and this is our present goal, mainly for Brownian motions on Riemannian manifolds. As we will see, there are numerous ways to construct such couplings (this is true in more general contexts, see [START_REF] Miclo | On the construction of measure-valued dual processes[END_REF] for the diversity of such couplings in a finite framework), but none of them is immediate and they are related to fine geometric features of the evolving subsets, such as their skeletons. We are thus to consider synchronous intertwined, free intertwined, mirror set-valued intertwined dual processes.

The reader must be warned that, as it stands now in the context of multidimensional diffusions, the set-valued dual processes are not defined up to the absorption time (except in symmetric settings), and as a consequence the same will be true for our couplings, which will be defined only up to some positive stopping times. We hope to investigate this point in future works, to end the construction of strong stationary times for Brownian motion on compact Riemannian manifolds, which remains our remote motivation. Other motivations for the couplings of primal and dual processes in the context of diffusions can be found in Machida [START_REF] Motoya Machida | Λ-linked coupling for drifting Brownian motions[END_REF] and [START_REF] Miclo | On the construction of measure-valued dual processes[END_REF].

Let us now present more precise definitions. Here the state space M is a d-dimensional complete Riemannian manifold. Denote respectively by ρ, µ and µ, the Riemannian distance, the Lebesgue measure on M and the corresponding pd ´1q-Hausdorff measure. The main objective of this paper is to construct couplings of primal diffusions processes with their set-valued dual intertwined processes. This will partially solve Conjecture 6 in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] in the case of Brownian motion pX t q tě0 and stochastic modified mean curvature flow pD t q tě0 (which were generically denoted p Xt q tě0 above). This conjecture says that an intertwined construction in the sense of Definition 1.1 is always possible. Definition 1.1. Consider a Markov process D " pD t q tPr0,τ s , with values in compact subsets of M and continuous with respect to the Hausdorff topology, and where τ is an a.s. positive stopping time in the filtration F D of D, serving as a lifetime for D. We say that a Brownian motion X " pX t q tě0 in M and D are intertwined when for all bounded F D -stopping time τ 1 smaller than τ , conditioned on F D τ 1 , X τ 1 has uniform law in D τ 1 (and in particular X τ 1 P D τ 1 ). More generally, for any F D -stopping time r τ smaller than τ , we say that X and D are r τ -intertwined when X and pD t q tPr0,r τ s are intertwined.

This is a generic definition, below stronger topologies on subsets of M will be considered. Note that the above lifetime is not necessary the explosion time, i.e. the exit time from all compact sets for the considered topology. In the infinite dimensional state space of D, compactness does not seem an appropriate notion.

Our main results are Theorems 2.8, 3.5 and 4.1 presenting such joint constructions of the primal Brownian motion pX t q tě0 and the dual domain-valued pD t q tě0 processes. The coupling of Theorem 2.8 consists in the infinite-dimensional stochastic differential equation (2.10), based on a function f : px, Dq Þ Ñ f px, Dq which is a deformation of the signed distance from x P M to the boundary of the domain D (see Assumption (2.2) for the precise requirements). Theorem 3.5 is obtained by specifying some approximating functions f . Given the trajectory pX t q tě0 of the Brownian motion, we construct the domain evolution pD t q tě0 using the local time of pX t q tě0 on the skeletons of pD t q tě0 and the mean curvatures of the normal foliations of these domains (see (3.30)). Other approximating functions f lead to Theorem 4.1, where the prominent role is played by the local time at the boundary. This situation is in some sense opposite to the previous one, since the driving Brownian motion of pD t q tě0 is now independent from pX t q tě0 , while it is as correlated as it can be in Theorem 3.5. These theoretical results are illustrated by the fundamental examples of Section 5. First we recover the intertwining relation between the real Brownian motion and the three-dimensional Bessel process. Next we deal with rotationally symmetric manifolds. Finally we present the application of our results to symmetric convex domains in the plane, even if the detailed proofs are deferred to a forthcoming paper.

To come back to our initial motivation, assume that X and D are intertwined, where the lifetime τ is the hitting/covering time by D of the whole state space M . If furthermore τ is finite (typically true when M is compact), then the Riemannian measure can be normalized into a probability (called the uniform distribution, which is invariant and reversible for the Brownian motion X) and τ is a strong stationary time for X. In this situation, the tail distributions of τ provide quantitative estimates for the speed of convergence of the Brownian motion toward equilibrium, in the separation sense. These estimates will need geometric ingredients such as Ricci bounds and it will be interesting to see how they will enter the game.

The needs for couplings between primal and dual processes of a Markovian intertwining relation is illustrated by [START_REF] Marc Arnaudon | On the separation cut-off phenomenon for Brownian motions on high dimensional spheres[END_REF], where strong stationary times τ n are constructed for the ndimensional sphere (when the subset-valued dual is starting from a singleton), satisfying Erτ n s " lnpnq n and for any r ą 0, For α P p0, 1q, denote by D 2`α the set of compact connected subsets D of M with C 2`α boundary. It will be more convenient to work with this state space (endowed with its natural topology) than with the larger one considered in Definition 1.1. Let us even restrict it further:

lim nÑ8 P " τ n ą p1 `rq lnpnq n  " lim nÑ8 P " τ n ă p1 ´rq lnpnq n  " 0.
We fix a point o P M for convenience.

Definition 2.1. For a given α P p0, 1q, ε ą 0, we denote by F α,ε the set of D P D 2`α such that ' D Ă Bpo, 1{εq the Riemannian ball centered at o with radius 1{ε; ' ρpBD, SpDqq ě ε, where S " SpDq is the skeleton of D (see appendix A for details); ' ρpBD, S out pDqq ě ε, where S out pDq is the outer skeleton of D, i.e. the skeleton of pDq c . ' the coefficients of the α-Hölderianity of the second fundamental form of BD are bounded by 1{ .

The set F α,ε will serve as the state space of the set-valued process p r D t q tPr0,τεs and τ ε P p0, `8s will be the exiting time from F α,ε . This process will be a diffusion, i.e. a Markov process with continuous trajectories (for the topology inherited from D 2`α ), and its generator Ă L will be defined later in (2.12). We extend the trajectory p r D t q tPr0,τεs by taking r D t " r D τε for any t ą τ ε . It amounts to imposing that Ă L vanishes outside F α,ε . It is possible to define in the same way p r D t q tPr0,τ q on D 2`α (which coincides with Y εą0 F α,ε ), where τ is the exiting time from D 2`α . But it will be more convenient for us to work with a process with an infinite lifetime (to be able to apply Proposition D.3 in Appendix D) and whose set of values has a boundary which is well-separated from the skeleton.

Let β P t0, αu. For D 0 P D 2`β and δ ą 0 small enough, a δ-neighborhood of D 0 is defined as follow:

V 2`β δ pD 0 q :" intpexp BD0 pf qq, f P C 2`β pBD 0 q, }f } C 2`β pBD0q ă δ ( ,
where for f P C 2`β pBD 0 q exp BD0 pf q :" exp x pf pxqN D0 pxqq, x P BD 0 ( (exp being the exponential map in M ), and intpexp BD0 pf qq is the interior of the hypersurface exp BD0 pf q, oriented by the orientation of D 0 . Let ηpBD 0 q ą 0 be the radius of the maximal tubular neighborhood of BD 0 . Notice that δ ă ηpBD 0 q garantees that all elements of V 2`β δ pD 0 q are regular deformations of D 0 . Also notice that all elements D of F α,ε have ηpBDq ě ε.

We identify two domains D 1 , D 2 P V 2`β δ pD 0 q with the functions f 1 , f 2 P C 2`β pBD 0 q such that D 1 " inttexp BD0 pf 1 qu and D 2 " inttexp BD0 pf 2 qu and we define a local distance

(2.2) d β,D0 pD 1 , D 2 q :" }f 1 ´f2 } C 2`β pBD0q . Assumption 2.2.
' The function

f : M ˆFα,ε Ñ R px, Dq Þ Ñ f px, Dq " f D pxq is a C 2`
σ c : M ˆFα,ε Ñ ΓpT M b pR m q ˚q px, Dq Þ Ñ σ c px, Dq " σ D
c pxq P LpR m , T x M q where ΓpT M b pR m q ˚q is the set of sections over M of T M b pR m q ˚and LpR m , T x M q is the set of linear maps from R m to T x M , such that the linear map

σ D pxq : R ˆRm Ñ T x M pw 0 , wq Þ Ñ w 0 ∇f D pxq `σD c pxqpwq (2.4) satisfies (2.5) @x P D, σ D pσ D q ˚pxq " Id TxM . Remark 2.3. The first condition of Assumption 2.2 implies that ∇f D | BD " p∇ρ BD q| BD p" N D q and ∆f D | BD " p∆ρ BD q| BD p" ´hD q. (2.6)
where h D stands for the mean curvature on BD. It also implies that the functions f D are uniformly Lipschitz and have uniformly bounded Laplacian. Also, for fixed x P BD, varying D successively along a field K normal to the boundary BD and along N D for the second derivative:

x∇f px, ¨q, Kypxq " ´xN D pxq, Kpxqy and

∇df px, ¨q `N D , N D ˘" 0 (2.7)
where ∇df px, ¨q is the Hessian of f in the second variable.

The second condition of Assumption 2.2 implies that for all u P T x M , (2.8)

}u} 2 " xu, ∇f D pxqy 2 `m ÿ i"1 xu, σ D c pxqpe i qy 2
for e 1 , . . . , e m an orthonormal basis of R m . In particular, if x P BD, taking u " ∇f D pxq " N D pxq, we get since }N D pxq} " 1:

(2.9) 0 " x∇f D pxq, σpxqpe i qy, i " 1, . . . m.

Proposition 2.4. Assumption 2.2 can always be realized, with any α P p0, 1q and ε ą 0.

Proof. We begin with remarking that for D P F α,ε , ρpBD, SpDqq ě ε. In particular, the distance to BD is C 2`α on D ε :" tx P M, ρpx, BDq ă εu. Let h ε be an odd smooth nondecreasing function from R to R `such that h ε prq " r for r P r0, ε{2s, h ε prq " p3{4qε for r ě ε and }h 1 ε } 8 ď 1. Then f D :" h ε ˝ρB D satisfies all the requirements of the first condition of Assumption 2.2. Then for constructing σ D c we proceed as in [START_REF] Marc Arnaudon | Reflected Brownian motion: selection, approximation and linearization[END_REF], Proposition 3.2 taking σ 1 " ∇f D . The wanted regularity in D is easily checked.

Let W t and W m t two independent Brownian motions with values respectively in R and R m .

The equation we are interested in writes in Itô form for all y P BD t :

(2.10)

# dX t " `∇f Dt pX t q dW t `σDt c pX t q dW m t dBD t pyq " N Dt pyq `dW t ``1 2 h
Dt pyq `∆f Dt pX t q ˘dt started at a compact domain D 0 with C 2`α boundary and X 0 such that L pX 0 q " U pD 0 q, where U pD 0 q is the uniform probability measure on D 0 . The notation dBD t pyq stands for an infinitesimal move of the boundary BD t at point y and is rigorously presented in Appendix B, see (B.7). In fact, as in Definition 2.1, the evolution equation (2.10) is implicitly considered only up to the exit time τ ε of F α,ε for some fixed α P p0, 1q, ε ą 0, after which the process is assumed not to move.

In (2.10), the processes pD t q tě0 and pX t q tě0 are fully interacting, since the evolution of one of them depends on the other one. In particular, they are not Markovian by themselves in general.

Another subset-valued process p r D t q tě0 will be interesting for our purposes. It is solution to the evolution equation

@ t ď r τ , @y P B r D t , dB r D t pyq " N r Dt pyq ˜dĂ W t `˜1 2 h r Dt pyq ´µB r Dt pB r D t q µp r D t q ¸dt ¸, (2.11)
where Ă W t is a real-valued Brownian motion and where r τ is the exit time from F α, . Notice that the equation for r D t does no longer depend of X t , so if the solution is unique, p r D t q tě0 will be Markovian. It is Equation (44) in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] (up to a time scaling by 2). Theorem 40 of [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] (where (44) has been rewritten as (79)) proves local existence of a solution.

Theorem 2.5. Fix α P p0, 1q and ε ą 0. Then (2.11) admits a unique global solution. In particular the process p r D t q tě0 is Markovian.

Proof. The proof is a consequence of Theorem 22 in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]. It can be found in Appendix C.

To describe the generator Ă L of p r D t q tě0 we must introduce the following notations. For any smooth function k on M , consider the mapping F k on D 2`α by

@ D P D 2`α , F k pDq :" ż D k dµ
For any k, g P C 8 pM q and any D P D 2`α , define

Ă L rF k spDq :" µ BD pkq µ BD pBDq µpDq ´1 2 µ BD px∇k, N D yq (2.12) Γ Ă L rF k , F g spDq :" ż BD k dµ ż BD g dµ (2.

13)

Next consider A the algebra consisting of the functionals of the form F :" fpF k1 , ..., F kn q, where n P Z `, k 1 , ..., k n P C 8 pM q and f : R Ñ R is a C 8 mapping, with R an open subset of R n containing the image of D 2`α by pF k1 , ..., F kn q. For such a functional F, define

Ă L rFs :" n ÿ l"1 B j fpF k1 , ..., F kn q Ă L rF k l s (2.14) `n ÿ j,lP 1,n B j,l fpF k1 , ..., F kn qΓ Ă L rF kj , F k l s
To two elements of A, F :" fpF k1 , ..., F kn q and G :" gpF g1 , ..., F gm q, we also associate

Γ Ă L rF, Gs :" ÿ lP n ,jP m B l fpF k1 , ..., F kn qB j gpF g1 , ..., F gm qΓ Ă L rF k l , F gj s (2.15)
Remark 2.6. To see that the above definitions are non-ambiguous, since a priori they could depend on the writing of F P A under the form fpF k1 , ..., F kn q and similarly for G, see Remark 2 of [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]. More generally, the forms of (2.14) and (2.15) are consequences of the diffusion feature of Ă L , for more on the subject, see e.g. the book of Bakry, Gentil and Ledoux [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF].

Remark 2.7. In the above considerations, Ă L was defined on D 2`α , but from now on, Ă L will stand for the restriction of this generator to F α,ε and will be zero on D 2`α zF α,ε , in accordance with Definition 2.1. Similarly, all stochastic differential equations will be valid only up to the stopping time τ ε (which was defined after Definition 2.1) or r τ ε (defined after (2.11)).

The interest of Assumption 2.2 comes from the following result:

Theorem 2.8. Let px, Dq Þ Ñ f D pxq and px, Dq Þ Ñ σ D
c pxq satisfy Assumption 2.2. Then equation (2.10) has a solution pX t , D t q tě0 started at D 0 P F α,ε , X 0 " U pD 0 q. Moreover the processes pX t q tě0 and pD t q tě0 are τ ε -intertwined.

Proof. We prove here the existence of solution to equation (2.10). The intertwining will be a consequence of Proposition 2.11 below.

We begin to prove the existence of a diffusion with modified drift, and then we will get the result by change of probability. It is possible to extend the description of L to more general functions on M ˆFα,ε (it vanishes on its complementary set), by replacing F k in (2.34) by a mapping F from A, as presented before Theorem 2.8.

Let pP t q tě0 be the Markovian semi-group associated to the processes pX t , D t q tě0 solution to (2.10) stopped at τ ε . This semi-group is associated to L in the weak sense of martingale problems, as described in Appendix D.

Let p r D t q tě0 be a diffusion process with generator Ă L stopped outside F α,ε , started at r D 0 " D 0 (due to Theorem 2.5, this process can be obtained as a solution to the evolution equation (2.11)), r ν t its law at time t and let (2.39) ν t pdD, dxq :" r ν t pdDqU pDqpdxq.

Proposition 2.11. We have for all smooth functions g, k on M :

(2.40) B t ν t pgF k q " ν t pL pgF k qq.

As a consequence, if pD 0 , X 0 q has law ν 0 then for all t ě 0, the solution pD t , X t q to equation (2.10) has law ν t , implying that pX t q tě0 and pD t q tě0 are τ ε -intertwined. Moreover D t is a diffusion with generator Ă L .

Proof. Integrating (2.34) in x with respect to the uniform law μD :" U pDq in D yields (2.41)

´μ D `g∆f D ˘µBD pkq´1 2 μD pgqµ BD px∇k, N D yq`1 2 F k pDqμ D p∆gq´µ BD pkqμ D px∇g, ∇f D yq. By Stokes theorem, (2.42) μD `g∆f D `x∇g, ∇f D y ˘" μBD `gx∇f D , ´N D y ˘" ´μ BD pgq, so the expression (2.41) writes (2.43) HpDq :" µ BD pkqμ BD pgq ´1 2 μD pgqµ BD px∇k, N D yq `1 2 F k pDqμ D p∆gq
On the other hand (2.44)

ν t pgF k q " r ν t rμ Dt rgsF k s which implies that (2.45) B t ν t pgF k q " B t r ν t p `μ Dt pgqF k ˘" r ν t ´Ă L `μ Dt pgqF k ˘¯. By (2.23), (2.46) Ă L `μ Dt pgq ˘" ´1 2 μBDt px∇g, N Dt yq, so, taking into account (2.13), Ă L `μ Dt pgqF k " μDt pgq Ă L pF k q `Fk Ă L `μ Dt pgq ˘`Γ Ă L " μDt pgq, F k ‰ " μDt pgq ! µ BDt pkqμ BDt pBD t q ´1 2 µ BDt px∇k, N Dt yq ) ´1 2 µ Dt pkqμ BDt px∇g, N Dt yq ´`´μ BDt pgq `μ Dt pgqμ BDt pBD t q ˘µBDt pkq " ´1 2 μDt pgqµ BDt px∇k, N Dt yq ´1 2 µ Dt pkqμ BDt px∇g, N Dt yq `μ BDt pgqµ BDt pkq.
But μDt p∆gq " ´μ BDt px∇g, N Dt yq and F k pD t q " µ Dt pkq, so (2.47)

HpD t q " Ă L `μ Dt pgqF k ˘,
which together with (2.45) proves (2.40).

Let us now prove that for any t ě 0, P t transports ν 0 into ν t , where pP t q tě0 is the semi-group introduced after the proof of Proposition 2.10. Consider the map (2.48) Gpg, k, tqpsq " ν s pP t´s pgF k qq , s P r0, ts.

We compute

Gpg, k, tq 1 psq " pB s ν s q pP t´s pgF k qq ´νs pB t P t´s pgF k qq " ν s pL P t´s pgF k qq ´νs pL P t´s pgF k qq " 0 (2.49) where we used Proposition D.3 in Appendix D to justify the differentiations (as well as the fact that L P t´s pgF k q " P t´s L pgF k q is bounded to be able to use differentiation under the integral ν s ). So we get Gpg, k, tqp0q " Gpg, k, tqptq which rewrites as (2.50) ν 0 P t pgF k q " ν t pgF k q, More generally, by similar arguments, we can replace in this formula F k by any mapping F from A. This in turn implies that ν 0 P t " ν t .

To finish, by iteration, we see that if X 0 " μD0 then pD t q tě0 has the same finite time marginals as p r D t q tě0 , proving that pD t q is a diffusion with generator Ă L .

INTERTWINED DUAL PROCESSES: A GENERALIZED PITMAN THEOREM

In this section we will consider the case where f D is the distance to boundary. It is not covered by Section 2 since distance to boundary is not smooth, it is singular on the skeleton of D. We will make an approximation of it, and then go to the limit in law.

Let Ă W t be a real-valued Brownian motion and r D t be the solution of (2.11) started at r D 0 , with driving Brownian motion Ă W t .

Assumption 3.1. Fix α P p0, 1q and ε ą 0. There exists a closed bounded subset r F α,ε of F α,ε in which the process p r D t q tě0 a.s. takes its values, such that the map D Þ Ñ SpDq is continuous from r F α,ε with the C 2 metric to KpM q, the set of compact subsets of M endowed with the Hausdorff metric. Moreover Brownian motions with probability one never hit the singular part of Sp Dt q. Conjecture 3.2. We conjecture that Assumption 3.1 is always realized, for any α P p0, 1q, ε ą 0, r D 0 P F α,ε .

Notice that Theorem 1.1 in [START_REF] Albano | On the stability of the cutlocus[END_REF] proves the first part of the conjecture, i.e. the continuity of D Þ Ñ SpDq, in the case where M " R d endowed with a possibly varying Riemannian metric. All examples together with the study of the motion of the skeleton in Appendix B make us believe that Conjecture 3.2 is true. However a better knowledge of skeletons is necessary to solve it. We believe that the process pSp Dt qq tě0 takes its values in a set of regular stratified spaces, and that it has absolutely continuous variation in this space.

Let us begin with some preparatory results. To describe the approximation of ρpx, BDq we are interested in, let us introduce some notations.

' Let px, Dq Þ Ñ ε px, Dq :" ph ε ˝ρBD qpxq where h ε " 1 in r0, ε{2s, h ε " 0 in r3ε{4, 8q and h ε is smooth and nonincreasing in r0, 8q. When D is fixed by the context, we will denote ε pxq :" ε px, Dq.

' For any δ P p0, εq, let ϕ δ : R `Ñ R be a nonnegative function with support in r0, δs, such that the mapping

R d Q u Þ Ñ ϕ δ p|u|q is smooth and ş R d ϕ δ p|u|q du " 1 (in
the sequel, | ¨| will stand for the usual Euclidean norm or for the Riemannian norm on any tangent space of M , depending on the context) .

' Let g δ be a smooth, 1-Lipschitz and odd function defined on R, with g δ prq " r on r0, ε{4s, 0 ď g δ prq ď r for any r ě 0, and g δ prq " c δ r on r3ε{8, 8q, for an appropriate constant c δ ď 1 very close to 1 that will be defined below in (3.2). We write ρ δ px, BDq :" g δ pρpx, BDqq.

The approximation of ρpx, BDq we choose is

f δ px, Dq " ε px, Dqρ δ px, BDq `p1 ´ ε px, Dqq ż TxM ϕ δ p|v|qρ δ pexp x pvq, BDq dv (3.1)
(where dv stands for the Lebesgue measure on T x M ). Define epδq :" supt~p∇ expqpuqs~, x P Bpo, 1{εq, u P B x p0, δq Ă T x M u where ∇ exppuq : T x M Ñ T exp x puq M is the covariant derivative of exp with respect to the base point, ~¨~is the operator norm, when T x M and T exp x puq M are endowed with their Euclidean structures, and B x p0, δq is the open ball in T x M with center 0 and radius δ. Recall that ε is fixed as in Assumption 3.1. The previously mentioned constant c δ is given by

c δ :" e ´1pδq p1 ´δ}∇ 1 ε } 8 q (3.2)
Notice that c δ does not depend on D and is as close as we want to 1. More precisely, we have Lemma 3.3. There exists two constants C 1 1 , C 2 1 ą 0, depending only on ε, such that for δ ą 0 sufficiently small,

0 ď epδq ´1 ď C 1 1 δ |c δ ´1| ď C 2 1 δ Proof.
The inequalities of the first line are well-known properties of the exponential mapping. The second bound follows, since }∇ 1 ε } 8 " }h 1 ε } 8 is independent of D (and of order 1{ε).

From the second bound, we can and will assume that the function g δ is furthermore chosen so that g δ prq converges uniformly to r on compact sets of R `, as well as the corresponding derivatives up to order 2 as δ OE 0. In addition, we choose δ ą 0 sufficiently small so that the map px, yq Þ Ñ exp ´1 x pyq is well-defined and smooth in the δ-neighborhood the diagonal of Bpo, 1{εq ˆBpo, 1{εq. Then, for any x P M , we can rewrite (3.1) under the form

f δ px, Dq " ε px, Dqρ δ px, BDq `p1 ´ ε px, Dqq ż M ϕ δ p| exp ´1 x pyq|qρ δ py, BDq J exp ´1 x pyqdy, (3.3)
where J exp ´1 x is the absolute value of the determinant of the Jacobian of exp ´1 x p¨q.

The interest of all these preparations is:

Proposition 3.4.
For all δ ą 0 sufficiently small, the function px, Dq Þ Ñ f δ px, Dq :" f D δ pxq has the following properties ' f δ satisfies the conditions of Assumption 2.2; ' there exists C 1 ą 0 such that @D P r F α,ε and x P D, we have

(3.4) |f δ px, Dq ´ρpx, BDq| ď C 1 δ;
' the differential and the Hessian of f δ with respect to the second variable D satisfy @D P r F α,ε , @x P DzSpDq, for all vector fields K normal to BD:

(3.5) xd 2 f δ px, Dq, Ky ď C 4 }K} 8 and }∇ 2 d 2 f δ px, Dq pN BD , N BD q} ď C 4
for a C 4 not depending on x, D, δ. The second term is the second derivative along the inward normal flow on D.

Proof. We first prove }d 1 f δ px, Dq} ď 1, d 1 denoting the differential with respect to the first or the x variable. For x P Bpo, 1{εq we have

d 1 f δ px, Dq " ε px, Dqd 1 ρ δ px, BDq `p1 ´ ε px, Dqqd 1 ˆżTxM ϕ δ p|u|qρ δ pexp x puq, BDq du ḋ1 ε px, Dq ż TxM ϕ δ p|u|q pρ δ px, BDq ´ρδ pexp x puq, BDqq du. (3.6) Notice that if x 1 is close to x and ı x,x 1 : T x M Ñ T x 1 M is the parallel transport along the minimal geodesic from x to x 1 , then ż T x 1 M ϕ δ p|u|qρ δ pexp x 1 puq, BDq du " ż TxM ϕ δ p|u|qρ δ pexp x 1 pı x,x 1 puq, BDq du.
Taking the differential with respect to x 1 at x 1 " x and using ∇ x 1 | x 1 "x ı x,x 1 " 0 by definition of parallel transport yields

d 1 ˆżTxM ϕ δ p|u|qρ δ pexp x puq, BDq du ˙" ż TxM ϕ δ p|u|qd 1 ρ δ pp∇ expqpuq, BDq du.
If ρpx, BDq ď ε{2 then ε px, Dq " 1, ∇ ε px, Dq " 0 and

}d 1 f δ px, Dq} ď ε px, Dq}d 1 ρ δ px, BDq} ď 1.
If ρpx, BDq ě ε{2 then for δ ď ε{8, we have, for u P T x M with |u| ď δ, ρpexp x puq, BDq ě 3ε{8. It follows

}d 1 f δ px, Dq} ď ε pxqe ´1pδq p1 ´δ}d 1 ε } 8 q `p1 ´ ε pxqq ż TxM ϕ δ p|u|qc d }p∇ expqpuq} du `}d 1 ε pxq} 8 ż TxM ϕ δ p|u|qδ du ď 1.
It is easily checked that the function f δ satisfies the other properties of Assumption 2.2.

Let us check that it also satisfies (3.4).

We have (3.7) for some constant C 3 1 ą 0 (depending on ε). This yields (3.4) with C 1 :" 1 `C3 1 . For proving (3.5), we take a vector field Kpyq " kpyqN pyq, y P BD and compute (3.8) xd 2 ρpx, BDq, Ky " x´N pP pxqq, KpP pxqqy " ´kpP pxqq where P pxq is the projection of x onto BD, and

f δ px, Dq´ρ δ px, BDq " p1´ ε px, Dqq
∇ 2 d 2 ρpx, BDq pN BD , N BD q " 0. (3.9)
Remarking that }d 2 ε px, Dq} is bounded by }h 1 ε } 8 , we get (3.5) via a straightforward computation.

Theorem 3.5. Fix D 0 " D0 P r F α,ε and let X 0 " U pD 0 q. Under Assumption 3.1, there exists a pair pX t , D t q tě0 of τ ε intertwined processes in the sense of Definition 1.1, such that the process pD t q tě0 satisfies dBD t pyq " N Dt pyq ˜@dX t , N Dt pX t q D `ˆ1 ∇ρ BD pxq " N D pxq and ∆ρ BD pxq " ´hD pxq and we will see that on the moving skeleton S t " S Dt :

(3.13) "∆ρ BDt pX t q dt" " ´2 sinpθ St pX t qq dL St t pXq. Proof. £Under Assumption 3.1, Proposition 3.4 allows us to construct for each δ ą 0, intertwined processes pX δ t , D δ t q tě0 started at pX δ 0 , D δ 0 q " pX 0 , D 0 q, associated with the functions f D δ , stopped at τ δ ε , the exit time from r F α,ε . We have from Equation (2.10)

dBD δ t pyq " N D δ t pyq ˆdW δ t `ˆ1 2 h D δ t pyq `∆f D δ t δ pX δ t q ˙dt (3.14)
for some Brownian motion W δ t . On the other hand, from Proposition 2.11 and (2.1), (3.15) p r D δ t q tě0 :" pD δ t q tě0 satisfies equation (2.11):

dBD δ t pyq " N D δ t pyq ˜dĂ W δ t `˜1 2 h D δ t pyq ´µBD δ t pBD δ t q µpD δ t q ¸dt (3.16)
where Ă W δ t is the

F D δ t ´Brownian motion (3.17) d Ă W δ t " dW δ t `∆f D δ t δ pX t q dt `µBD δ t pBD δ t q µpD δ t q dt.
A remarkable fact about all pX δ t , D δ t q tě0 is that their marginals are constant in law. Notice that also ppD δ t q tě0 , τ δ ε q is constant in law since τ δ ε is a functional of pD δ t q tě0 independent of δ. As a consequence, the family

(3.18) ´pX δ t , D δ t , W δ t , Ă W δ t , W δ,m t q tě0 ,
E " F g pD δ t q F 1 pD δ t q
F k pD δ t q  and passing to the limit yields the intertwining.

This property of pD δ t , Ă W δ t q tě0 being constant in law passes to the limit, and we have

dBD t pyq " N Dt pyq ˆdĂ W t `ˆ1 2 h Dt pyq ´µBDt pBD t q µpD t q ˙dt ˙. (3.20)
We need to work with real-valued processes: we have from (2.32), for all δ ą 0,

(3.21) ż t 0 dµpD δ s q µpBD δ s q " ´W δ t ´ż t 0 ∆ 1 f δ pX δ s , D δ s q ds.
This together with (3.17) yields

dBD δ t pyq " N D δ t pyq ˆ´dµpD δ s q µpBD δ s q `1 2 h D δ t pyq dt (3.22)
Again by constantness in law:

dBD t pyq " N Dt pyq ˆ´dµpD s q µpBD s q `1 2 h Dt pyq dt ˙. (3.23)
So to prove our result we only need to prove that (3.24)

ż t 0 dµpD s q µpBD s q " ´Wt `ż t 0 h Ds pX s q ds `ż t 0
2 sin `θSs pX s q ˘dL Ss s pXq and that

(3.25) W t " ż t 0 xN Ds pX s q, dX s y.
Let us prove (3.25). In all this paragraph we consider M as isometrically embedded in some Euclidean space. In particular we are allowed to integrate vectorial quantities. We use the fact that dX δ t bdW δ t converges in law to dX t bdW t (where b stands for bracket of semimartingales). But dX δ t b dW δ t is equal to ∇ 1 f δ pX δ t , D δ t q dt. Then by Lemma G.1 applied to ∇ 1 f δ pX δ t , D δ t q (which is uniformly bounded) and U " tpx, Dq, x R SpDqu defined in (G.3) we see that the integral of ∇ 1 f δ pX δ t , D δ t q dt converges to the one of N Dt pX t q dt. But almost surely N Dt pX t q has norm 1 dt-a.e., implying that dW t " xN Dt pX t q, dX t y.

Let us now establish (3.24). It will be a consequence of the convergence of pf δ pX δ t , D δ t qq tě0 to pρpX t , BD t q tě0 . Write the Itô formula for f δ pX δ t , D δ t q:

d `fδ pX δ t , D δ t q ˘"xd 1 f δ pX δ t , D δ t q, dX δ t y `1 2 ∆ 1 f δ pX δ t , D δ t q dt `xd 2 f δ pX δ t , D δ t q, dBD δ t y `1 2 ∇ 2 d 2 f δ pX δ t , D δ t qpdBD δ t , dBD δ t q dt `x∇ 2 d 1 f δ pX δ t , D δ t q, dBD δ t b dX δ t y. ( 3
ˆż t 0 xd 1 f δ pX δ s , D δ s q, dX δ s y ˙tě0 L ÝÑ ˆż t 0 xd 1 ρpX s , BD s q, dX s y ˙tě0 .
More precisely, we have a sequence of martingales converging in law to a martingale M t which is a Brownian motion by Theorem 3 in [START_REF] Wei | Tightness results for laws of diffusion processes; application to stochastic mechanics[END_REF]. For identifying the limiting martingale we use the convergence of xd 1 f δ pX δ s , D δ s q, dX δ s y b dX δ s to dM s b dX s obtained again by Theorem 3 in [START_REF] Wei | Tightness results for laws of diffusion processes; application to stochastic mechanics[END_REF] (here again we use an isometric embedding of M ). But Lemma G.1 proves that the limit is equal to ∇ 1 ρpX s , BD s q ds, yielding (3.29).

Next we prove that (3.30)

ˆż t 0 xd 2 f δ pX δ s , D δ s q, dBD δ s y ˙tě0 L ÝÑ ˆż t 0 xd 2 ρpX s , BD s q, dBD s y ˙tě0 .
The argument is similar except that as we see with (3.14), the drift part of dBD δ s is not well controlled as X δ t approaches the skeleton. So one cannot proceed exactly the same way. But fortunately, for x outside a 3ε{4-neighbourhood of BD and outside SpDq, we have

xd 2 f δ px, Dq, N | BD y " c δ ż TxM ϕ δ p|u|qx´N pP pexp x puqq , N pP pexp x puqqy du " ´cδ (3.31)
where c δ is defined in (3.2). This together with (3.22) suggests to write

ż t 0 xd 2 f δ pX δ s , D δ s q, dBD δ s y " ˆż t 0 xd 2 f δ pX δ s , D δ s q, dBD δ s y `cδ ż t 0 xN D δ s , dBD δ s y ċδ ż t 0 xN D δ s , dBD δ s y.
The second line clearly converges. The right hand side in the first line can be written

(3.32) ż t 0 ˜ ε pX δ s , D δ s q A d 2 f δ pX δ s , D δ s q `cδ N D δ s , dBD δ s E
with px, Dq Þ Ñ ˜ ε px, Dq :" p hε ˝ρBD qpxq where hε " 1 in r0, 3ε{4s, hε " 0 in rε, 8q and hε is smooth and nonincreasing in r0, 8q.

With this last integral we can proceed as for (3.29), after passing to the limit, and since lim δÑ0 c δ " 1, we get (3.31).

Similarly we obtain the two following convergences for the second derivatives.

ˆż t 0 ∇ 2 d 2 f δ pX δ s , D δ s qpdBD δ s , dBD δ s q ˙tě0 L ÝÑ ˆż t 0 ∇ 2 d 2 ρpX s , BD s q `N pP BDs pX s q, N pP BDs pX s q ˘ds ˙tě0 " 0 (3.33)
where P BDs pX s q is the orthogonal projection of X s on BD s (which is defined ds-almost everywhere), In this section we consider another canonical and extremal situation, the case where f D vanishes almost everywhere. More precisely, it is the limiting situation where f D is constant outside a ε-neighbourhood of the boundary. This situation is completely opposite to the one of Section 3 where the coupling is maximal.

ˆż t 0 x∇ 2 d 1 f δ pX δ s , D δ s q, dBD δ t b
Theorem 4.1. There exists a pair pX t , D t q tě0 of τ ε -intertwined processes in the sense of Definition 1.1 satisfying 4.1) can be considered as a limiting case of (2.10). Here Assumption 3.1 is not needed since the morphological skeleton of D does not play a role, and the map D Þ Ñ BD is already sufficiently regular.

(
Proof. The proof is quite similar to the one of Theorem 3.5, but with another family of functions f D δ , namely f D δ :" h δ ˝ρBD where h δ is defined in the proof of Proposition 2.4: h δ is a smooth nondecreasing function from r0, 8q to R `such that h δ prq " r for r P r0, δ{2s, h δ prq " p3{4qδ for r ě δ and }h 1 δ } 8 ď 1. But here, as ε is fixed, we will let δ OE 0. Again we construct for each δ ą 0, an intertwined processes pX δ t , D δ t q tě0 stopped at τ δ ε . Again all pX δ t , D δ t q tě0 are tight, and a limiting process pX t , D t q tě0 stopped at τ ε provides an intertwining. The proof of (4.1) goes along the same lines as the one of (3.10).

We end this section with another canonical construction, where the functions f D δ approximate ´ρBD . Theorem 4.3. Under assumption 3.4, there exists an intertwining pX t , D t q tě0 stopped at τ ε , satisfying

dBD t pyq "N Dt pyq ˜´@ dX t , N Dt pX t q D `ˆ1 2 h Dt pyq `hDt pX t q1 DtzSt pX t q ˙dt `2 sinpθ St pX t qq dL St t pXq ´2dL BDt t pXq (4.2)
Proof. It is completely similar to the ones of Theorems 3.5 and 4.1.

SOME FUNDAMENTAL EXAMPLES

5.1. Real Brownian motion and three-dimensional Bessel process. We come back to the case where M " R. Assume that the Brownian motion X starts from 0 (to respect rigorously the above framework, X should start from the uniform distribution on D 0 :" r´ , s and next we should let go to 0 `). Due to the invariance by symmetry of (3.10), for any t ą 0, D t remains a symmetric interval, let us write it r´R t , R t s. In this simple setting, we have N Dt p¨q " ´signp¨q on Rzt0u, h Dt " 0 and S t " t0u, for any t ą 0. Thus (3.10) writes

(5.1) dR t " signpX t qdX t `2dL t where L :" pL t q tě0 is the local time of X at 0. Namely we get that

@ t ě 0, R t " ż t 0 signpX s q dX s `2L t " |X t | `Lt
by Tanaka's formula. It is well-known that R :" pR t q tě0 is a Bessel process of dimension 3 (cf. e.g. Corollary 3.8 of Chapter 6 of Revuz and Yor [START_REF] Marc Yor | Grundlehren der Mathematischen Wissenschaften[END_REF]). In particular, we get that with the notation introduced in (A.4), @ t ě 0, ρ BDt pX t q " minpX t `Rt , R t ´Xt q But except at time t " 0, this quantity is always positive: a.s. X t never touch the boundary of D t for t ą 0. Indeed, if for some t ą 0 we have |X t | " R t , we deduce that L t " 0, namely a contradiction, since X 0 " 0.

In particular, we see that the intertwining coupling we have constructed is different from the one proposed by Pitman [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF], which is a.s. touching (the upper) boundary repeatedly. Instead we end up with the intertwining dual constructed in [START_REF] Miclo | On the construction of measure-valued dual processes[END_REF] via stochastic flows. It is mentioned there how to deduce the classical Pitman's dual, via Lévy's theorem.

Here is an alternative approach. While Equation (5.1) is obtained from approximating x Þ Ñ |r ´x| outside an ε-neighbourhood of 0 when D " r´r, rs by smooth functions f D satisfying Assumption 2.2, we are able to recover Pitman theorem by rather approximating x Þ Ñ ´x in D " r´r, rs outside the only ε-neighbourhood of ´r. In the limit of (2.10) as goes to zero, on the one hand we have (5.2)

1 tXt "Rtu dR t " dX t ,
on the other hand we have X t `Rt ě 0, so that X t `Rt is the solution to the Skorohod problem associated to 2X t . We get

(5.3) R t `Xt " 2X t ´2 min 0ďsďt X s .
which is equivalent to

(5.4) R t " X t ´2 min 0ďsďt X s .
The answer to the question: what would be a symmetric construction with local time at the two ends of r´R t , R t s is given by Theorem 4.3. We obtained intertwined processes with (5.5) R t " ´ż t 0 signpX s q dX s ´2L 0 t pXq `2L 0 t pR ´Xq `2L 0 t pR `Xq. We will investigate set-valued processes D t " Bpo, R t q where Bpo, rq is the open geodesic ball centered at o, with radius r. The skeleton of Bpo, R t q is the point o.

Let X t be a Brownian motion in M satisfying X 0 " U pD 0 q for some D 0 " Bpo, r 0 q. Denote by ρ t :" rpX t q the radial part of X t . Then (5.8) dρ t " dβ t `1 2 bpρ t q dt, ρ 0 " U f pp0, r 0 qq where pβ t q tě0 is a real Brownian motion and

(5.9) U f pdrq :" f prq ş r0 0 f psq ds dr.

The evolution equation (3.10) for D t shows by symmetry that for all t ě 0, D t " Bp0, R t q for some real-valued process pR t q t . Moreover it writes dρ t " dβ t `1 2 bpρ t q dt dR t " dβ t `"´1 2 bpR t q `bpρ t q  dt.

(5.10)

Proposition 5.1. The system of equations (5.10) has a solution up to explosion time of pR t q t

(5.11)

τ D :" inftt ě 0, R t R p0, 8qu,
which satisfies for all t ă τ D , (5.12)

0 ă ρ t ă R t .
The corresponding set-valued process D t " Bpo, R t q is solution to equation (3.10), and in particular, for all F D -stopping time τ , (5.13) L pX τ |F D τ q " U pD τ q as well as L pρ τ |F D τ q " U f pp0, R τ qq. Proof. We only have to check (5.12). By (5.10), (5.14) dpR t ´ρt q " 1 2 rbpρ t q ´bpR t qs dt, which vanishes on tR t " ρ t u, and since b is smooth, if ρ 0 ă R 0 , then ρ t ă R t for all times.

5.3. Brownian motion and annulus in 2-dimensional rotationally symmetric manifolds. Let M be a complete 2-dimensional Riemannian manifold, rotationally symmetric around a point o P M . Denote by pr, θq polar coordinates with rpxq " ρpo, xq and

(5.15)

ds 2 " dr 2 `f 2 prq dθ 2
the metric in polar coordinates. Then the radial Laplacian is

(5.16) ∆ r " B 2 pBrq 2 `bprq B Br with b " pln f q 1 .
If 0 ď r ´ď r `, let (5.17) Apr ´, r `q :" tx P M, r ´ď rpxq ď r `u if r ´ă r `, Apr ´, r `q :" H, the closed annulus delimited by the radius r ´and r `.

In the following we will investigate set-valued processes D t " ApR t , R t q. The skeleton of ApR t , R t q is the circle (5.18) S t " Cpo, R 0 t q with R 0 t :"

1 2 pR t `Rt q.
Let X t be a Brownian motion in M satisfying X 0 " U pD 0 q for some D 0 " Apr 0 , r 0 q. Denote by ρ t :" rpX t q the radial part of X t . Then (5.19) dρ t " dβ t `1 2 bpρ t q dt, ρ 0 " U f ppr 0 , r 0 qq where β t is a real Brownian motion and

(5.20)

U f ppr 0 , r 0 qqpdrq :" f prq ş r 0 r 0 f psq ds dr.
The evolution equation (3.10) for D t shows by symmetry that for all t ě 0, D t " ApR t , R t q for some real-valued processes R t ď R t . Moreover it writes

dρ t " signpρ t ´R0 t q dW t `1 2 bpρ t q dt dR t " dW t `"´1 2 bpR t q `signpρ t ´R0 t qbpρ t q  dt `2L R 0 t t pρq dR t " ´dW t `"´1 2 bpR t q ´signpρ t ´R0 t qbpρ t q  dt ´2L R 0 t t pρq R 0 t " 1 2 `Rt `Rt (5.21)
and these equations imply (5.22) dR 0 t " ´1 4 " bpR t q `bpR t q ‰ dt.

Proposition 5.2. The system of equations (5.21) has a solution up to explosion time (5.23) τ D :" inftt ě 0, pR t , R t q R p0, 8q 2 u, which satisfies for all t ă τ D ,

(5.24) R t ď ρ t ď R t .
The corresponding set-valued process D t " ApR t , R t q is solution to equation (3.10), and in particular, for all F D -stopping time τ , (5.25) L pX τ |F D τ q " U pD τ q as well as L pρ τ |F D τ q " U f ppR τ , R τ qq.

Proof. Fix ε ą 0 and α P p0, 1q. We will first solve the system of equations until the exit time τ ε and then let ε OE 0. Let us construct functions f D δ pxq which satisfies equation (3.1). It will be easier here because there is no need of functions ε and g δ .

For δ P p0, εq, let ϕ δ : R Ñ R be the function with support equal to r´δ{2, δ{2s, satisfying for ´δ{2 ă r ă δ{2: (5.26)

ϕ δ prq :" 1 cpδq exp ˜´1 `δ 2 ˘2 ´r2 ¸with cpδq :" ż δ{2 ´δ{2 exp ˜´1 `δ 2 ˘2 ´s2 ¸ds,
and let sign δ : R Ñ R r Þ Ñ ´1 `2 ż r ´8 ϕ δ psq ds.

(5.27)

The functions ϕ δ and sign δ are both smooth and Lipschitz, and they respectively approximate δ 0 and sign. For 0 ă r ´ă r `satisfying r `´r ´ě 2ε, defining r 0 :" 1 2 pr ´`r `q, for x P Apr ´, r `q let (5.28)

f Apr ´,r `q pxq " f px, r ´, r `q " gprpxqq with gprq " gpr, r ´, r `q " ż r r ´´sign δ ps´r 0 q ds.

Clearly f px, r ´, r `q is 1-Lipschitz in the first variable. A computation shows that (5.29) B r `gpr, r ´, r `q " ż r 0 ´ε ϕ δ pvq dv and B r ´gpr, r ´, r `q " ´ż ε r´r 0

ϕ δ pvq dv
showing that g and f are 1-Lipschitz. Then the vector N :" N BApr ´,r `q is equal to ´1trpxq"r `u B r ``1 trpxq"r ´uB r ´so that (5.30) x∇f, N y " 1 and ∇df pN, N q " 0.

This yields an elementary proof of the properties of Proposition 3.4. We can use Theorem 3.5 to solve equation (5.21) until the stopping time τ ε .

We are left to prove that τ ε Õ τ D a.s. as ε OE 0. This is a direct consequence of the fact that the volume of ApR t , R t q is a time changed Bessel process of dimension 3 (by [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] Theorem 5), proving that ApR t , R t q cannot collapse onto its skeleton.

Remark 5.3. After the hitting time of 0 by R t , the processes can continue to evolve under the regime of Section 5.2.

We recover from Proposition 5.2 a result from [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF] stating that prR t , R t sq tě0 is an intertwining dual process for the real diffusion pρ t q tě0 . In particular, we deduce that if pρ t q tě0 is positive recurrent and if `8 is an entrance boundary, then prR t , R t sq tě0 reaches r0, `8s in finite time and this finite time is a strong stationary time for pρ t q tě0 , see [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF] for more details. 5.4. Brownian motion and symmetric convex sets in R 2 . In this section we take M " R 2 endowed with the Euclidean metric. For any integer n ě 2, let G n the group of isometries of R 2 generated by the rotation of angle 2π n and the symmetry with respect to the horizontal axis. Consider a smooth strictly convex bounded set D 0 Ă M with smooth boundary, stable by the action of G n . Also assume that its skeleton has the form S 0 " G n H 0 , H 0 being an horizontal interval H 0 " r0, x 0 s ˆt0u for some x 0 ą 0. An example of such a set when n " 2 is the interior of an ellipse, the skeleton being the interval between the two foci. Assume that X t is a Brownian motion in R 2 satisfying X 0 " U pD 0 q. Let us investigate the evolution of pX t , D t q. Notice that it is the first example where we really have to deal with infinite dimensional processes. By conservation of the convexity by the normal and mean curvature flows, D t will stay convex. It will also stay symmetric. All the results of this subsection will be proved in the forthcoming paper [START_REF] Marc Arnaudon | Intertwining Brownian motions with symmetric convex sets[END_REF]: Proposition 5.4. The skeleton of D t always takes the form S t " G n H t with H t " r0, x t st 0u an horizontal interval.

Proof. See [START_REF] Marc Arnaudon | Intertwining Brownian motions with symmetric convex sets[END_REF] Denote by pı, q the canonical basis of R 2 , and X t " pX p1q t , X p2q t q. In this notation, when ´π{n ă θ St pX t q ă π{n, the vector N Dt pX t q of Equation ( 3 takes its values in a closed subset r F α,ε of F α,ε , invariant by G n , such that on r F α,ε , the map D Þ Ñ h D | BD is continuous from r F α,ε (with the C 2`α metric) to C 2 pBDq. Its skeleton St satisfies St " G n pr0, xt s ˆt0uq for some process xt . Proof. See [START_REF] Marc Arnaudon | Intertwining Brownian motions with symmetric convex sets[END_REF] In the next result we prove that the skeleton has finite variation and is monotonly decreasing.

Proposition 5.6. The right endpoint px t , 0q in the horizontal axis of the skeleton St satisfies

(5.33) dx t dt " ρ 2 ppx t , 0q, ỹt q 2 ph Dt q 2 pỹ t q,
ỹt being the point of B Dt in the horizontal line with the greatest abscissa, and the second derivative being calculated with curvilinear coordinates on B Dt . Notice that ph Dt q 2 pỹ t q ď 0, proving that the process Sp Dt q is monotonly decreasing.

Proof. Let us investigate the motion of a point in St close to px t , 0q. This point has two closest points in B Dt , which we call r y 1,t and r y 2,t , the first one having positive second coordinate. We will use Theorem B.1 and (B.28). Call xt the point in the skeleton corresponding to r y 1,t and r y 2,t . We have N 1 px t q " ´cos θpx t qı ´sin θpx t q, N 2 px t q " ´cos θpx t qı `sin θpx t q, N S 1 px t q " ´. Denote T pr y 1,t q the tangent vector to B Dt at r y 1,t , corresponding to increasing of θ: T pr y 1,t q " ´sin θpx t qı `cos θpx t q. Write h 1 pr y 1,t q the curvilinear derivative of hpr y 1,t q in the direction of T pr y 1,t q. Then the vector J K 1 p1q

of (B.28) is equal to ´1 2 ρ S pr y 1,t qh 1 pr y 1,t qT pr y 1,t q. So we get from (B.28):

d dt xt " 1 2
ρ S pr y 1,t qh 1 pr y 1,t q ˆsin θpx t q `cos 2 θpx t q sin θpx t q ˙ı " ρ S pr y 1,t qh 1 pr y 1,t q 2 sin θpx t q ı " ρ 2 S pr y 1,t qh 1 pr y 1,t q 2r y p2q 1,t ı with r y 1,t " pr y p1q 1,t , r y p2q 1,t q.

(5.34)

In the limit, as r y p2q 1,t goes to zero, we obtain the motion of xt and using the symmetry of the convex set, we have h 1 pr y t q " 0 so that we can replace h 1 pr y 1,t q r y p2q 1,t by h 2 pr y t q. This yields (5.33).

In particular a Brownian motion X t will never meet the ends of St . A solution to (5.32) can be found with the help of Theorem 3.5. The family of functions f δ px, Dq defined in (3.1) takes the form:

f δ px, Dq " ε pxqρ δ px, BDq `p1 ´ ε pxqq ż R 2 ϕ δ p|x ´y|qρ δ py, BDq dy " ε pxqρ δ px, BDq `p1 ´ ε pxqq ż R 2
ϕ δ p|y|qρ δ px ´y, BDq dy.

(5.35)

The investigation of the lifetime of the solution to (5.32) is not easy. In [START_REF] Marc Arnaudon | Intertwining Brownian motions with symmetric convex sets[END_REF] we prove that the lifetime is the time when D t meets its skeleton S t . So it is enough to investigate the time τ when Dt meets its skeleton St . We have no example where this happens. The next proposition yields examples where the lifetime is infinite, together with nice properties related to the symmetry group G n . Proposition 5.7.

(1) the process ˜µB r Dt pB r D t q µp r D t q ¸0ďtăτ is a supermartingale;

(2) when S0 is G n -symmetric with n ě 3, then the entropy process ´Ą Ent t ¯0ďtăτ defined as the integral of ρ log ρ with respect to the curvilinear abscissa in B Dt , ρ being the curvature of B Dt , is a supermartingale; (3) when S0 is G n -symmetric with n ě 7, then τ " 8 a.s. Consequently, when S 0 is G n -symmetric with n ě 7, Equation (5.32) provides an intertwining with infinite lifetime.

Proof. See [2] APPENDIX A. AN INTEGRATION BY PARTS ON DOMAINS WITH BOUNDARY

Our goal here is to obtain an extension of Stokes's formula on a domain with a smooth boundary, for functions which degenerate on the skeleton. We take the opportunity to recall this notion, as well as related geometric concepts.

Let M be a d-dimensional Riemannian manifold and D Ă M a compact and connected domain with smooth boundary BD. For y P BD, let N pyq be the inward normal vector. Denote by S 1 the inward (morphological) skeleton of D: S 1 is the set of points in D such that (i) the distance to BD is not smooth and (ii) there are points around them where the distance to BD is smooth with a non vanishing gradient. Denote (A. [START_REF] Albano | On the stability of the cutlocus[END_REF] τ pyq " inftt ą 0, exp y ptN pyqq P S 1 u.

Let S be the set of regular points of S 1 , which we can describe as follows: if x P S, then there exists a unique couple py 1 , y 2 q of distinct points from BD such that (A.2)

x " exp y1 pτ py 1 qN py 1 qq " exp y2 pτ py 2 qN py 2 qq .

We have τ py 1 q " τ py 2 q, and for i " 1, 2, the differential at pτ py i q, y i q of the map R `BD Q pt, yq Þ Ñ exp y ptN pyqq is nondegenerate. The set S is a codimension 1 submanifold of M and S 1 zS has Hausdorff dimension smaller than or equal to d´2. It is the union of the focal set which is the set of points x " exp y pτ pyqN pyqq such that pt, y 1 q Þ Ñ exp y 1 ptN py 1 qq is degenerate at pτ pyq, yq, and the union of the sets defined like S but withstrictly more than two points y 1 , y 2 , y 3 ,... For r ě 0, let

(A.3)
Dprq " tz P DzS 1 , ρ BD pzq ě ru.

where ρ is the Riemannian distance. The set Dprq is a (possibly empty) manifold with smooth boundary BDprq on which one can define an inward normal N pyq and an orientation by parallel transporting oriented basis of BD along normal geodesics. So we have for all y P DzS 1 : N pyq " ∇ρ BD pyq.

We will also need the sets Dprq for all r P R. We will let for r ă 0 and ψptq " ψp0, tq. We will indifferentely write ψptqpxq " ψpt, xq. The function ψps, tq is not defined for all points of BDpsq because we ask ψps, tqpyq P BDptq, nor is N p¨q. However for |s| and |t| small it is a map, defined for all y P BDpsq, and is is also a diffeomorphism with inverse ψpt, sq.

We have for 0 ď s ď t, ψptq " ψps, tq ˝ψpsq, which implies (A.6) det T ψptq " det T ψps, tq ˆdet T ψpsq.

Notice that thanks to the orientation of the sets BDprq we get an orientation of DzS 1 by adding N as first vector to oriented basis, consequently det T ψ is well defined and always positive. It is well-known that

(A.7) d dt ˇˇt "s det T ψps, tqpyq " ´hpyq
where hpyq is the inward mean curvature of BDpsq (the minus sign of the r.h.s. of (A.7) insures that h is non-negative on BDpsq when Dpsq is convex). This together with (A.6) yields Applying this formula to the function gh which we assume to be bounded below or integrable, we get by integration by parts For z " ψpτ py i q, y i q P S (i " 1, 2) define θpzqP p0, π{2s the angle between N pψpτ py i q´, y i qq and S. In the sequel we assume that θpzq " π{2 (the case θpzq " π{2 is simpler to deal with and Proposition A.1 is always valid). Notice that this angle does not depend on i, this is a consequence of z P S staying at the same distance to y 1 and y 2 by infinitesimal variation. For later use, let also θpzq " 0 when z P S 1 zS. Let us prove that for z " ψpτ py i q, y i qq P S, (A. [START_REF] Zheng | Tightness criteria for laws of semimartingales[END_REF] det T ψpτ py i q, y i q " sin θpϕpy i qq det T ϕpy i q, i " 1, 2.

Set y " y 1 . Let e 1 " N pyq, e S 1 " N pψpτ pyq´, yqq, N S pzq the normal to S at z such that xN S pzq, e S 1 y ą 0, let e 2 " pe 3 , . . . , e d q be a family of orthonormal normalized vectors in T y BD such that letting e 2 " ∇τ pyq }∇τ pyq} (we have ∇τ pyq " 0, since θpzq " π{2), e 1 :" pe 2 , e 2 q is an orthonormal basis of T y BD, let pe S q 2 " pe S 3 , . . . , e S d q be an orthonormal basis of T y ϕpVectpe 2 qq, let e S 2 such that pe S q 1 :" pe S 2 , . . . , e S d q is an orthonormal basis of T z S. Finally let e θ 2 P T z M be such that xe θ 2 , N pzqy ă 0 (e θ 2 and N S pzq are not orthogonal, since θpzq " π{2) and pe S

1 , e θ 2 , pe S q 2 q is an orthonormal basis of T z M . Figure 1 shows the configuration of e S 1 , N S pzq, e S 2 and e θ 2 on an example of dimension 2. In the sequel we will denote for instance T ϕpe 1 q "

¨T ϕpe 2 q . . .

T ϕpe d q

‹ ', so that xT ϕpe 1 q, pe S q 1 y will be the matrix of all scalar products. We have xT ϕpe 1 q, pe S q 1 y " xdτ, e 1 yxB t ψpτ pyq, yq, pe S q 1 y `xT ψpe 1 q, pe S q 1 y " ˆxdτ, e 2 yxB t ψ, e S 2 y `xT ψpe 2 q, e S 2 y xT ψpe 2 q, pe S q 2 y xdτ, e 2 yxB t ψ, e S 2 y `xT ψpe 2 q, e S 2 y xT ψpe 2 q, pe S q 2 y ˙.

Let us simplify and make more explicit this expression. We have xdτ, e 2 y " 0. Also e θ 2 K pe S q 2 and e S 2 K pe S q 2 so e S 2 P Vectpe S 1 , e θ 2 q and more precisely (A.17) e S 2 " cospθpzqqe S 1 `sinpθpzqqe θ 2 .

On the other hand T ψpe 1 q K e S 1 which implies xT ψpe 1 q, e S 2 y " sinpθpzqqxT ψpe 1 q, e θ 2 y.

Also xB t ψ, e S 2 y " cospθpzqq. We arrive at detxT ϕpe 1 q, pe S q 1 y " sin θpzq det ˆxT ψpe 2 q, e θ 2 y xT ψpe 2 q, e θ 2 y xT ψpe 2 q, pe S q 2 y xT ψpe 2 q, pe S q 2 y ċos θpzq det ˆxdτ, e 2 y 0 xT ψpe 2 q, pe S q 2 y xT ψpe 2 q, pe S q 2 y " sin θpzq det T ψ `cos θpzqxdτ, e 2 y detxT ψpe 2 q, pe S q 2 y.

(A. [START_REF] Kaj Nystr | The Skorohod oblique reflection problem in time-dependent domains[END_REF] For the last equation we used the fact that det T ψ " detxT ψpe 1 q, pe θ 2 , pe S q 2 qy, since e 1 and pe θ 2 , pe S q 2 q are orthonormal bases. Note that by definition, xT ψpe 2 q, e θ 2 y " 0, so we also get det T ψ " detxT ψpe 2 q, pe S q 2 y ˆxT ψpe 2 q, e θ 2 y. On the other hand, we have

(A.20)
xdτ, e 2 y " xT ψpe 2 q, e θ 2 y cot θpzq.

Indeed, note that

0 " @ T ϕpe 2 q, N S D " xdτ, e 2 y @ e S 1 , N S D `@T ψpe 2 q, N S D " xdτ, e 2 y sinpθpzqq ´cospθpzqq @ T ψpe 2 q, e θ 2 D
where the last term is obtained by taking into account that T ψpe 2 q is parallel to e θ 2 . This is the change of length of the geodesic needed to stay in S. We obtain det T ϕ " sin θpzq det T ψ `cos θpzq cot θpzq det T ψ " sin 2 θpzq `cos 2 θpzq sin θpzq det T ψ.

This yields (A.16). We arrived at Using the change of variable y Þ Ñ ϕpyq and the fact that all z P S is equal to ϕpy i q, i " 1, 2, we obtain the key formula Proposition A.1. With the above notations, for any smooth function g defined on D such that gh is integrable or bounded below, we have: In this section we describe how to move a domain with smooth boundary by deformation of its boundary. We will investigate the deformation of its skeleton The deformation we will consider will have a general absolutely continuous finite variation part, together with a very specific martingale part and singular finite variation part. First we introduce some notation.

For a domain D with smooth boundary BD, s P R, define

ψ D psq " ψ D p0, sq : BD Ñ BDpsq y Þ Ñ ψ D psqpyq " ψ D ps, yq " exp y `sN D pyq ˘. (B.1)
Here N D " N is the inward normal defined in Section A. Consider a moving domain t Þ Ñ D t . Be careful not to confound Dptq with D t , since in general they are quite different subsets. We first assume that the deformation is sufficiently regular so that for all 0 ď s ď t, we can write D t as (B.2) D t " ! ψ Ds prZ Ds t pyq, τ Ds pyqs, yq, y P BD s

) .

In particular, we must have S 1 s Ă D t . Notice that in the special case where the real valued function t Þ Ñ Z Ds t pyq does not depend on y, for any 0 ď s ď t, then we have

(B.3) D t " D s pZ Ds t q " D 0 pZ D0 t q, Z D0 t " Z Ds t `ZD0 s
where Dprq is defined in (A.3), replacing distance to BD by signed distance with positive sign inside D and negative sign outside. In this situation, the skeleton is not moving, at least as long as BD t remains smooth (i.e. until BD t hits S 1 0 or is too far outside D 0 ), and t Þ Ñ Z D0 t can be allowed to be a semimartingale with singular continuous drift. When t Þ Ñ Z Ds t pyq depends on y the situation is a little bit more complicated. Starting from pt, yq Þ Ñ Z D0 t pyq which is assumed to be defined on r0, εq ˆBD 0 , the sets D t are defined for 0 ď t ă ε, as well as the Z Ds t pyq, 0 ď s ď t, y P D s . In fact, if py, tq Þ Ñ Z D0 t pyq is C 1 , then one can reconstruct all Z Ds t pyq with the only knowledge of 9 Z Dt t pzq, z P BD t . Let us do it for s " 0: the map pt, yq Þ Ñ ψ D0 pt, yq from p´α, αq ˆBD 0 to M is a diffeomorphism on its range, for α ą 0 sufficiently small. Let us denote z Þ Ñ pτ 0 pzq, ϕ 0 pzqq its inverse. Then a variation z `N Dt pzqdZ Dt t corresponds to a variation pτ 0 pzq, ϕ 0 pzqq `pdτ 0 , T ϕ 0 qN Dt pzqdZ Dt t of the coordinates in p´α, αq ˆBD 0 . But this is not convenient at all, since it is not intrinsic. Moreover, when passing to stochastic processes and Stratonovich equations, it will involve second derivatives of z Þ Ñ pτ 0 pzq, ϕ 0 pzqq. So we prefer to leave the reference to D 0 and to always stay at the level of the moving D t .

For all y P BD 0 we define a stochastic process t Þ Ñ Y t pyq representing the motion of D t satisfying Y 0 pyq " y and the Itô equation in manifold with respect to the Levi Civita connection ∇ (B.4) dY t pyq " d ∇ Y t pyq " B 1 ψ Dt p¨, Y t pyqqpdZ Dt t pY t pyqqq " N Dt pY t pyqqdZ Dt t pY t pyqq. Recall that fomally d ∇ Y t pyq is a vector which writes in local coordinates py 1 , . . . , y d q with the Christoffel symbols Γ i j,k :

(B.5) d ∇ Y t pyq " ˆdY i t pyq `1 2 Γ i j,k pY t pyqq dxY j t pyq, Y k t pyqy ˙Di pY t pyqq
where D i pY t pyqq is the vector B By i taken at point Y t pyq. We will always assume that the martingale part dm t of dZ Dt t pyq does not depend on y. In this situation, the Itô equation is equivalent to the Stratonovich one: indeed, using (B.3) the Itô to Stratonovich convertion term is

1 2 ∇ N D t pYtpyqqdmt N Dt p¨qdm t " 1 2 ∇ N D t pYtpyqq N Dt p¨qdxm, my t " 0
since N Dt pY t pyqq is the speed at time a " 0 of the geodesic a Þ Ñ ψ Dt paqpY t pyqq.

More precisely, we will let dZ Dt t pyq be of the form

(B.6) dZ Dt t pyq " H Dt pY t pyqq dt `dz t
where H Dt is a smooth function on BD t (which later on will be chosen to be h Dt {2, where h Dt is the mean curvature of BD t ) and pz t q tě0 is a real valued continuous semimartingale. We assume that Equation (B.4) has a strong solution up to some positive stopping time. Moreover, since dY t pyq represents the motion of BD t and for small time the map y 1 Þ Ñ Y t py 1 q is a diffeomorphism from BD 0 to BD t , writing Y t py 1 q " y, equation (B.4) rewrites as (B.7) dBD t pyq:"dY t py 1 q " N Dt pyq `HDt pyq dt `dz t ˘.

Let us now investigate the motion of the skeleton S t under this motion of D t . First we remark that by local inversion theorem, at regular points of the skeleton, the variation in Stratonovich sense is linear and the sum of all variations of the concerned point at the boundary. As we already remarked, the motion dz t does not change S t , so this together with the linearity just mentioned implies that we have a finite variation of the skeleton.

Recall the situation of (A.2) in Section A. We consider a domain D, x P S, y 1 , y 2 the two elements of BD such that exp y1 pτ py 1 qN py 1 qq " exp y2 pτ py 2 qN py 2 qq, with τ py 1 q " τ py 2 q. For i " 1, 2, we will consider a variation of the minimal geodesic from y i to x, represented by a Jacobi field J i satisfying J i p0q P T yi M , J 1 p1q " J 2 p1q P T x M , (B.8) J i p0q " λ i N py i q `JK i p0q, J 1 i p0q " λ 1 i N py i q `pJ K i q 1 p0q, with J K i orthogonal to N py i q. The motion of S corresponding to the motion of y 1 and y 2 will be represented by J 1 p1q. Since S has a boundary, the observation of the orthogonal part to S of J 1 p1q is not sufficient.

Let γ i be the projection on M of J i . It is the geodesic in time 1 from y i to x (as usual in the computations of Jacobi fields, the speed is not normalized). Denote N i pxq " 9 γ i p1q{} 9 γ i p1q}. Recall that the angle between N i pxq and T x S is θpxq P p0, π{2s. We will also let (B.9)

N S 1 pxq " 1 2 sin θpxq pN 1 pxq ´N2 pxqq.

Figure 2 shows the configuration of the points x, y 1 , y 2 and the vectors N 1 pxq, N 2 pxq, N S 1 pxq. The vector N S 1 pxq is is the normal vector to S at point x, in the same side as N 1 pxq. We will consider variations of geodesics with same final value:

(B.10) J 1 p1q " J 2 p1q " λN S 1 pxq `JT 1 p1q FIGURE 2.
The points x, y 1 , y 2 and the vectors N 1 pxq, N 2 pxq, N S 1 pxq

for some λ P R, where J T 1 p1q P T x S. Writing λN S 1 pxq " Next we will compute the tangential displacement J T p1q of x in S. As we will see later, we will only need a Jacobi field J 1 such that J K 1 p0q and pJ K 1 q 1 p0q are known and (B.16) J 1 p0q " λ 1 N py 1 q, i.e. J K 1 p0q " 0.

So we know J K 1 p1q: and (B.17)

J K 1 p1q " J `1, 0, pJ K 1 q 1 p0q
where Jp1, u, vq is the value at time 1 of the Jacobi field J with Jp0q " u and J 1 p0q " v. From

J 1 p1q " J T 1 p1q `xJ 1 p1q, N S 1 pxqyN S 1 pxq J 1 p1q " J K 1 p1q `xJ 1 p1q, N 1 pxqyN 1 pxq (B.18)
we get

(B.19) J T 1 p1q " J K 1 p1q `xJ 1 p1q, N 1 pxqyN 1 pxq ´xJ 1 p1q, N S 1 pxqyN S 1 pxq.
On the other hand we have

xJ 1 p1q, N 2 pxqy " xJ K 1 p1q, N 2 pxqy `xJ 1 p1q, N 1 pxqyxN 1 pxq, N 2 pxqy xJ 1 p1q, N 2 pxqy " xJ 1 p1q, N 1 pxqy ´pλ 1 ´λ2 q (B.20)
where the second equation is a direct consequence of (B.15). Substracting the second equation to the first one yields (B. [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF] p1 ´cosp2θpxqqqxJ 1 p1q, N 1 pxqy " xJ K 1 p1q, N 2 pxqy `λ1 ´λ2 . Replacing xJ 1 p1q, N 1 pxqy in (B. [START_REF] Kaj Nystr | The Skorohod oblique reflection problem in time-dependent domains[END_REF]) and after simplification, using (B.9) and (B.15), we finally obtain the horizontal displacement (B.22)

pJ T 1 qp1q " J K 1 p1q`1 4 sin 2 θpxq `2xJ K 1 p1q, N 2 pxqyN 1 pxq `pλ 1 ´λ2 qpN 1 pxq `N2 pxqq ˘.
We are now in position to write the motion of the skeleton S t when the motion of the boundary is given by (B.7). For x P S t with corresponding points y 1 and y 2 in BD t , (B.23) dS K t pxq " 1 2 sin θ St pxq `HDt py 1 q ´HDt py 2 q ˘N St 1 pxq dt which has finite variation. Observe that, as already mentioned, the term dz t disappears.

Here we wrote dS K t pxq for the normal variation of the regular skeleton. But as we already remarked, since S t is not a closed manifold, it can expand via the motion of its boundary. So we have to investigate the horizontal motion dS T pxq.

Notice that J K 1 q 1 p0q is the perpendicular part of the time derivative of the speed at y 1 of the geodesic in time 1 from y 1 to x. So from equation (B.7) we deduce the rotation (B. [START_REF] Ren | Brownian motion[END_REF] pJ K 1 q 1 p0q dt " ρ S py 1 q∇ t N Dt py 1 q " ´ρS py 1 q∇H Dt py 1 q dt. (in the r.h.s. the gradient corresponds to the tangential gradient on BD t , recall that H Dt is only defined on this hypersurface).

We conclude that the horizontal displacement of x is J T 1 p1q dt where p S denotes the orthogonal projection on T S, J K 1 p1q " Jp1, 0, ´ρS py 1 q∇H Dt py 1 qq, and y 1 , y 2 are defined in Figure 2.

J T 1 p1q dt " J K 1 p1q
Remark B.2. The points y 1 and y 2 do not play the same role in Theorem B.1. As formula (B.27) is symmetric in y 1 and y 2 , formula (B.28) is not. The reason is that if we assume the motion of y 1 to be normal to the boundary BD t and to have speed given by (B.26), the motion of y 2 has no reason to be normal to the boundary: J K 2 p0q does not vanish.

APPENDIX C. DOSS-SUSSMAN REPRESENTATION OF IT Ô'S EQUATION (2.11) In this section we adapt the results of [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] to our notations. Let the stochastic mean curvature flow be a solution of : @ t P r0, τ q, @ y P C t , dBD t pyq " ˆdW t `1 2 h Dt pyqdt ˙N Dt pyq (C. [START_REF] Albano | On the stability of the cutlocus[END_REF] where C t :" BD t , starting at D 0 .

Let BG t be a solution of " G 0 " D 0 @ t P r0, r q, @ x P BG t , B t x " α BGt,´Wt pxqN Gt pxq (C.2) for some r ą 0 small enough, where α is defined by @ r ą 0, @ D P D r , @ x P C, α C,r pxq :" 1 2 h ΨpC,rq pψ C,r pxqq (C.3) and ΨpC, rq is the normal (exterior) flow starting at C at time r (c.f. Chapter 3 and 4 of [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] for notations).

Similarly to the proof of Theorem 17 from [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF], we show that D t " ΨpG t , ´Wt q is a solution of the stopped martingale problem associated to the generator pD, r

Lq where for f P C 8 pM q and F f pDq " ş D f dµ, ν " ´N is the exterior normal r LF f pDq :" 1 2 ż BD x∇f, νy dµ " F 1 2 ∆f pDq. Recall that the equation (C.2), is in fact a quasiparabolic equation with coefficients that depend on trajectory of the Brownian motion (the meaning is trajectory by trajectory). Similarly to Section 4.1 from [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF], we show that the solution of (C.2) have a regularity C 1`α 2 ,2`α , for all α ă 1.

Proposition C.1. Let BG t be a solution of (C.2). Then BD t " ΨpBG t , ´Wt q is a solution of (C.1) in the Itô sense.

Proof. Let x P ΨpBG t , ´Wt q, we have : By the uniqueness of the solution of (C.2) (c.f. Theorem 22 in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]) and the fact that it is adapted to the filtration of B we deduce that the solution of (C.5) is unique and is a strong solution. Similarly we have the uniqueness of the solution of dBD t pxq " ˆdW t `1 2 h BDt pxqdt ´µpBD t q µpD t q dt ˙N BDt pxq.

dΨpBG t ,
Moreover, since we could also make a change of time in the Itô equation, Equation (2.11) has a unique strong solution.

APPENDIX D. WEAK SEMI-GROUP THEORY IN THE MARTINGALE PROBLEM SENSE

This theory has been developed in several books, see for instance Stroock and Varadhan [START_REF] Stroock | Multidimensional diffusion processes[END_REF] or Ethier and Kurtz [START_REF] Ethier | Markov processes[END_REF]. Here we present a minimal version suitable for our purposes.

Let V be a measurable state space and consider Ω a set of trajectories from R `to V . The canonical coordinates on Ω are denoted by the X t , for t ě 0: for ω P Ω, X t pωq is the position at time t of ω. The set Ω is endowed with the sigma-field generated by the X t , for t ě 0. Our first assumption is that the mapping Ω ˆR`Q pω, tq Þ Ñ X t pωq P V is measurable, which usually means that "Ω is not too big".

For t ě 0, we define F t :" σpX s : s P r0, tsq

For t ě 0, we will also need the time shift Θ t associating to any ω P Ω the trajectory Θ t pωq defined by @ s ě 0, X s pΘ t pωqq " X s`t pωq

We assume that Θ t pΩq Ă Ω.

A given family P :" pP x q xPV of probability measures on Ω is said to be Markovian if for any x P V and any t ě 0, the image by Θ t of P x conditioned by F t is P Xt . In particular, it is assumed that P has the regularity of a Markov kernel from V to Ω.

From now on, we suppose that a Markovian family P is given. Let B be the space of bounded and measurable functions defined on V . The semi-group P :" pP t q tě0 associated to P is the family of operators acting on B via @ t ě 0, @ f P B, @ x P V, P t rf spxq :" E x rf pX t qs

The Markovianity of P implies at once the semi-group property @ s, t ě 0, P t P s " P t`s and in particular the elements of P commute. A subclass of "regular" functions that will be important for our purposes is R defined as R :" " f P B : @ x P V, lim tÑ0`P t rf spxq " f pxq * Exceptionally in the above limit, we assumed that t ě 0 (i.e. not only that t ą 0), so that by definition, for any f P R and x P V , P 0 rf spxq " f pxq.

Let us observe that R is left stable by the semi-group:

Lemma D.1. For any t ě 0, we have P t rRs Ă R. Thus for any given f P R and x P V , the mapping

R `Q t Þ Ñ P t rf spxq is right continuous.
Proof. Indeed, fix t ě 0 and f P R, we have for any x P V and s ě 0, P s rP t rf sspxq " P t rP s rf sspxq " E x rP s rf spX t qss

We have for any s ě 0, }P s rf s} 8 ď }f } 8 (where }¨} 8 stands for the supremum norm on B) and since f P R, we get everywhere lim sÑ0`P s rf spX t q " f pX t q Dominated convergence implies that lim sÑ0`E x rP s rf spX t qss " E x rf pX t qs " P t rf s as desired.

The generator L associated to P is the operator L : DpLq Ñ R defined in the following way: the space DpLq is the set of functions f P R for which there exists a function g P R such that the process M f,g :" pM f,g t q tě0 defined by @ t ě 0, M f,g t :" f pX t q ´f pX 0 q ´ż t 0 gpX s q ds is a martingale under P x , for all x P V . Let us remark that g is then uniquely determined. Indeed, we have for any x P V and t ě 0, (we came back to the usual convention that t ą 0 in the above limit) and as a by-product, we are assured of the existence of the latter limit. We define Lrf s :" g and M f :" M f,g . The differentiation property (D.1) can be extended into Lemma D.2. For any f P DpLq, x P V and t ě 0, we have B t P t rf spxq " P t rLrf sspxq (D.2) Proof. For any f P DpLq, x P V and t, s ě 0, we have

E x rf
E x " M f t`s ´M f t ı " E x " E x " M f t`s ´M f t |F t ıı " 0
We compute that We can now come to the main goal of this appendix: Proposition D.3. For any t ě 0, DpLq is stable by P t and on DpLq we have LP t " P t L.

M f t`s ´M f t " f pX t`s q ´f pX t q ´ż t`s t Lrf spX u q du so that E x " M f t`s
Proof. Fix f P DpLq and x P V , the assertion of the lemma amounts to checking that the process N :" pN s q sě0 defined by pN s q sě0 :" ˆPt rf spX s q ´Pt rf spX 0 q ´ż s 0 P t rLrf sspX u q du ˙sě0 is a martingale under P x . Consider s 1 ě s ě 0, we have to prove that

E x rN s 1 ´Ns |F s s " 0 (D.3)
The l.h.s. is equal to E x « P t rf spX s 1 q ´Pt rf spX s q ´ż s 1 s P t rLrf sspX u q du ˇˇFs ff " E x « P t rf spX s 1 ´s ˝Θs q ´Pt rf spX 0 ˝Θs q ´ż s 1 ´s 0 P t rLrf sspX u ˝Θs q du ˇˇFs ff " E y « P t rf spX s 1 ´sq ´Pt rf spX 0 q ´ż s 1 ´s 0 P t rLrf sspX u q du ff where y " X s . By Fubini's lemma, the previous r.h.s. can be written The advantage of the above approach is that it is quite sable by optional stopping, as it is the case for martingales. Let us succinctly give a simple example in the spirit of Section 2.

E y rP t rf
Assume that in the above framework, V is a metric space, endowed with its Borelian measurable structure, and that Ω is the set of continuous trajectories CpR `, V q. Furthermore, we suppose that P is Fellerian, in the sense that it preserves C b pV q, the set of bounded and continuous real functions on V .

Let be given A Ă V a closed set. We consider τ the hitting time of A: τ :" inftt ě 0 : X t P Au P R `\ t`8u

Define the "new" process r X :" p r X t q tě0 via @ t ě 0, r X t :" X t^τ and for x P V , let r P x be the image of P x by r X, it is still a probability measure on CpR `, V q. All notions corresponding to r P :" p r P x q xPV , which is still a Markovian family, receive a tilde. It appears without difficulty that r R is the set of functions r f P B such that there exists f P R with r f coinciding with f on V zA. The domain Dp r Lq is the set of r f P r R such that there exists f P DpLq with r f coinciding with f on V zA. In addition, we have

@ x P V, r Lr r f spxq " " Lrf spxq , when x R A 0 , when x P A
This expression does not depend on the choice of f , due to the fact that P is a diffusion, i.e. that Ω " CpR `, V q, which implies that L is a local operator (see for instance Theorem 7.29 of Schilling and Partzsch [START_REF] Ren | Brownian motion[END_REF], they are working with Euclidean spaces, but the result can be extended to metric spaces).

According to (D.2) and Proposition D.3, we get @ r f P Dp r Lq, @ x P V, @ t ě 0 B t r P t r r f spxq " r P t r r Lr r f sspxq " r Lr r P t r r f sspxq Such relations are not so obvious if we had chosen to work in a Banach setting (cf. e.g. the book of Yosida [START_REF] Ōsaku Yosida | Functional analysis[END_REF]), considering for instance semi-groups acting on the space C b pV q (endowed with the supremum norm), since in general r L would not naturally take values in C b pV q.

APPENDIX E. AN IT Ô-TANAKA FORMULA

Let M be a d-dimensional Riemannian manifold and D Ă M a compact and connected domain with C 2 boundary BD, and S be the regular skeleton of D, and ρ BD the signed distance to BD, which is positive inside D and negative outside D. The notations will be the same as in Appendix A.

Proposition E.1. Let X t a Brownian motion in M . We have the following Itô-Tanaka formula : dρ BD pX t q " xN D pX t q, dX t y ´1 2 h D pX t qdt ´sin `θS pX t q ˘dL S t pXq, in the above formula, N D pxq " ∇ρ BD pxq and ´hD pxq " ∆ρ BD pxq for x R S , and define to be 0 elsewhere, L S t pXq is the local time defined as in (3.11).

Proof. The formula is a consequence of the Itô formula outside the skeleton. Since the non regular part of the skeleton has Hausdorff dimension smaller than or equal to d ´2, it is not visited by the Brownian motion. So we only focus on the regular skeleton. For all x P S, the distance to the boundary is the minimum of two C 2 functions f, g defined on some neighborhood U of x in M . The function f (resp. g ) is the distance function to a piece of BD containing y 1 (resp. y 2 ) as in (A.2). We have locally,

ρ BD " f ^g " 1 2 pf `gq ´1 2 |f ´g|.
Using Itô formula and Tanaka formula we have Hence dL 0 t ppf ´gqpX . qq " 2 sin `θS pX t q ˘LS t pX . q. Together with (E.1), this yield the Proposition.

dρ BD pX t q " 1 2 ´1

APPENDIX F. UNIQUENESS IN LAW OF L DIFFUSION

Let us consider the following generator x L of a stochastic modified mean curvature flow. The action of this generator and its carré du champs on elementary observables are defined as follows. For any smooth function k on M , consider the mapping F k on D 2`α defined by

@ D P D 2`α , F k pDq :" ż D k dµ
For any k, g P C 8 pM q and any D P D 2`α ,

# x L rF k spDq :" ´1 2 µ BD px∇k, N D yq " F 1 2 ∆k pDq Γ x L rF k , F g spDq :" ş BD k dµ ş BD g dµ. (F.1)

Note that x

L has the same carré du champs as the carré du champs associated to Ă L . From now the generator x L is defined as in (2.14).

Proposition F.1. The martingale problem associated x L is well-posed.

Proof. We have already shown the existence result in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF], so it remains to prove the uniqueness in law. Let us first consider the two-dimensional Euclidean case, namely M " R 2 . For all λ P R and for any function k λ P vectpe λx , e λy q we have 1 2 ∆k λ px, yq " λ 2 2 k λ px, yq.

Let f λ ppx, yq, Dq :" k λ px, yqF k λ pDq, for px, yq P R 2 and D P D 2`α . This function satisfies the following property:

x L f λ ppx, yq, Dq " k λ px, yq x L F k λ pDq " k λ px, yqF 1 2 ∆k λ pDq " k λ px, yqF λ 2 2 k λ pDq " λ 2 2 k λ px, yqF k λ pDq " 1 2 ∆k λ px, yqF k λ pDq. " 1 2 ∆f λ ppx, yq, Dq
Let pX t q tě0 be a R 2 -valued Brownian motion that starts at X 0 " px 1 , x 2 q P R 2 and p Dt q tě0 a x L diffusion that starts at D 0 independent of pX t q tě0 . Even if we stop the diffusion, we can assume that its lifetime is infinite and we add indicators as described in Appendix D. For all 0 ď s ď t , we have

df λ pX t´s , Ds q m " ´1 2 ∆f λ pX t´s , Ds qds `x L f λ pX t´s , Ds qds m " 0.
Hence for all λ P R we have (F.2) Erf λ pX t , D 0 qs " Erf λ pX 0 , Dt qs.

Since the left hand side of the above equation does not depend on the x L diffusion, we get that for any x L diffusion p Dt q tě0 that starts at D 0 :

Erf λ pX 0 , Dt qs " Erf λ pX 0 , Dt qs, and so

ErF k λ pD t qs " ErF k λ p Dt qqs. In order to apply Theorem 4.2 of [START_REF] Ethier | Markov processes[END_REF], we have to show that the above equation characterizes the law of the one-dimensional distribution, i.e. we have to show that pF k λ q is separating in the space of probability measures on D 2`α . This is equivalent to separate domains. Let A, B P D 2`α such that F k λ pAq " F k λ pBq for all λ P R and k λ P xe λx , e λy y, we have for all λ:

ż A k λ px, yqdµ " ż B k λ px, yqdµ.
After successive derivations in λ and evaluation at λ " 0, we get for all n P N ż The above computations could be done also for kλ1,λ2 " e λ1x`λ2y , since 1 2 ∆ kλ1,λ2 "

λ 2 1 `λ2 2 2
kλ1,λ2 , and after derivations in λ 1 , λ 2 and evaluating at p0, 0q we get that for all n, m P N: ż A

x n y m dµ " ż B

x n y m dµ, hence, using the boundary regularity, we get A " B.

We could also apply Stone-Weierstrass' theorem to the function algebra generated by the mappings px, yq Þ Ñ e λ1x and px, yq Þ Ñ e λ2y .

The proof is the same for all Euclidean spaces. If M is a compact manifold let f λi pX, Dq :" k λi pXqF k λ i pDq,

where λ i is an eigenvalue of 1 2 ∆ and k i is the associated eigenfunction (respectively the Neumann eigenvalue). By the same computation as above (F.2) is also valid for the boundary reflecting Brownian motion), to get the conclusion we have to show that pF k λ i q i separates domains. Since pk λi q i is an orthonormal basis of L 2 pµq we get that if A, B P D 2`α be such that for all i, F k λ i pAq " F k λ i pBq i.e x1 A , k λi y L 2 " x1 B , k λi y L 2 , then 1 A L 2 " 1 B hence A " B. For the complete manifold M , let Ω k be an exhaustion of M with a regular boundary such that D 0 Ă Ω k , and stop the x L diffusion when it hit Ω c k and use the above result for the manifold with boundary Ω k , we get the result by localization.

Proposition F.2. The martingale problem associated to L is well-posed.

Proof. Let D t be a L diffusion that starts at D 0 , defined on pΩ, F D , Qq. We first recall that there exist an enlargement of the probability space such that it carries a one dimensional Brownian motion B such that for all k P C 8 pM q (F.3) F k pD t q " F k pD 0 q `ż t 0 L rF k spD s q ds `ż t 0 a Γ L rF k , F k spD s q dB s where a Γ L rF k , F k spDq :" ş BD k dσ, this is actually Proposition 53 in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]. Note that this procedure of enlargement (Theorem 1.7 chapter V in [START_REF] Marc Yor | Grundlehren der Mathematischen Wissenschaften[END_REF]) could be done by gluing the same independent Brownian motion for each pΩ, F D , Qq. We denote by p Ω, FD , Qq the enlarged probability space. Since L is an h-transform of x L namely Using Girsanov transform, D t is solution of the x L martingale problem on the probability space p Ω, FD , Pq. Since Q " M ´1P we get the uniqueness in law of the L diffusion by Proposition F.1.

L rF k s " x L rF k s `Γx L pF 1 , F k q F 1 ,

APPENDIX G. CONVERGENCE IN LAW: A KEY LEMMA

This Appendix is devoted to the adaptation to some domain-valued sequences of processes, of Lemma 4 in [START_REF] Wei | Tightness results for laws of diffusion processes; application to stochastic mechanics[END_REF], which states stability of some time integrals under convergence in law.

Lemma G.1. Let F :" F α,ε . We endow the set of continuous paths C ´r0, 8q, M ˆF with the two dissimilarity measures d β , β P t0, αu, defined as:

(G.1)
d β `px 1 , D 1 q, px 2 , D 2 q ˘" sup Here HpD, D 1 q is the Hausdorff distance between D and D 1 and the distance d β,D is defined in (2.2).

Let pX n t , D n t , τ n ε q tě0 :" pX δn t , D δn t , τ δn ε q tě0 a subsequence of (3.18) converging in law to the limit defined in (3.19) for the product of d α and the Euclidean distance in R `.

Let Proof. We will follow the proof of Lemma 4 in [START_REF] Wei | Tightness results for laws of diffusion processes; application to stochastic mechanics[END_REF], but with several differences due to infinite dimensional spaces. Set for n P N, t ě 0, (G.4) A n t :" ż t 0 f n pX n s , D n s q ds, A t :"

ż t 0 f pX s , D s q ds.
Condition (i) implies that the processes A n are tight. To get the conclusion il is sufficient to show that all the converging subsequences have the same limit. So assume that (G.5) pX n t , D n t , A n t q tě0 L ÝÑ pX t , D t , a t q tě0 . and let us prove that pa t q tě0 " pA t q tě0 . By Skorohod theorem we may realize all processes (G.6) pX n t , D n t , A n t , X t , D t , a t q tě0 on the same probability space pΩ, F , Pq in such a way that (G.7) pZ n t q tě0 :" pX n t , D n t , A n t q tě0 a.s. ÝÑ pX t , D t , a t q tě0 ": pZ t q tě0 . This means that Z n t Ñ Z t a.s. uniformly in t ě 0. Fix ω P Ω. Let t ą 0 be such that pX t pωq, D t pωqq P U . For some ε 1 ą 0 we have pX s pωq, D s pωqq P U for all s P rt ´ε1 , t `ε1 s. The set ) .

for some small enough ε 2 ą 0. For n sufficiently large, pX n s pωq, D n s pωqq P V for all s P rt´ε 1 , t`ε 1 s. On the other hand V is bounded for the distance d α . This implies by Arzela-Ascoli theorem that it is compact for the distance d 0 . We have the two following facts, the first one being an assumption on the f n and f , the second one being a consequence of the d 0 -compactness of V (a) f n Ñ f as n Ñ 8 uniformly in pV, d 0 q; (b) f is uniformly continuous in pV, d 0 q. Both terms in the right converge to 0, the first one by (a) and the second one by (b). So we have by (G.7) and the above calculation (G.10) " pA n s pωqq sPrt´ε,t`εs Ñ pa s pωqq sPrt´ε,t`εs ppA n s pωqq 1 " f n pX n s pωq, D n s pωqqq sPrt´ε,t`εs Ñ pf pX s pωq, D s pωqqq sPrt´ε,t`εs both uniformly in s P rt ´ε, t `εs. This implies that a s pωq is differentiable in pt ´ε, t `εq with derivative f pX s pωq, D s pωqq and in particular at t.

We have that for all t ě 0, pX t pωq, D t pωqq P U a.s.. So for all t ě 0, (G.11) d dt a t pωq " f pX t pωq, D t pωqq a.s.. This implies that ω a.s.

(G.12) d dt a t pωq " f pX t pωq, D t pωqq for a.e. t.

On the other hand we know by [START_REF] Zheng | Tightness criteria for laws of semimartingales[END_REF] Theorem 10 that pa t q tě0 is absolutely continuous : This together with (G.4) proves the lemma.

(G.

2 .

 2 INTERTWINED DUAL PROCESSES: EXISTENCE IN CONNECTION WITH STOKE'S FORMULA In this section we make a construction of intertwined processes X and D based on the Stokes' Formula (2.1) below. Consider a compact domain D in M with C 2 boundary. Let f : D Ñ R a C 2 function such that ∇f | BD " N D the normal inward vector on boundary. Then by Stoke's formula, for any C 2 function g : D Ñ R, (2.1) ´żBD gdµ " ż BD gx∇f, ´N D y dµ " ż D g∆f dµ `żD x∇g, ∇f y dµ.

  α function in the two variables (the differential in D is in the sense of Fréchet with respect to the above local Banach structure defined by the distances d α,D ). The functions f D satisfy (2.3) › › ∇f D › › 8 ď 1, and coincide with the signed distance to the boundary ρ BD (positive inside D and negative outside) in a neighbourhood of BD. The functions f D have bounded Hessian, uniformly in D P F α,ε . Furthermore, we assume that the coefficients of the α-Hölderianity of Hessf D are uniformly bounded over F α,ε . ' There exists a positive integer m and a C 1 map

ż

  TxM ϕ δ p|u|q pρ δ pexp x puq, BDq ´ρδ px, BDqq du which implies |f δ px, Dq ´ρδ px, BDq| ď δ. On the other hand |ρpx, BDq ´ρδ px, BDq| ď p1 ´cδ q max ˆ2 ,

5. 2 .

 2 Brownian motion and disks in rotationally symmetric manifolds. This is the simpler example since the skeleton is never hit by the Brownian motion. Consider a complete d-dimensional manifold with d ě 2, rotationally symmetric around a point o P M . Denote by pr, Θq polar coordinates with rpxq " ρpo, xq and (5.6) ds 2 " dr 2 `f 2 prq dΘ 2 the metric in polar coordinates. Then the radial Laplacian is (5.7) ∆ r " B 2 pBrq 2 `bprq B Br with b " pd ´1qpln f q 1 .

(A. 4 )

 4 Dprq " tz P M, ρ BD pzq ě ru where ρ BD is the signed distance to BD, positive inside D, negative outside D. Define for s, t P R ψps, tq : BDpsq Ñ BDptq y Þ Ñ exp y ppt ´sqN pyqq (A.5)

  " ´h pψpsqpyqq det T ψpsqpyq and consequently, using ψp0q " id and det T ψp0q " 1, (A.9) det T ψptqpyq " expˆż t 0 ´h pψpsqpyqq ds ˙.Denote by µ the volume measure of D and by µ the volume measures of the manifolds BDpsq and of S.with convention det T ψprqpyq " 0 if r ě τ pyq. ds ˙dr ¸µpdyq.More generally, for a measurable function g : D Ñ R bounded below, ds ˙dr ¸µpdyq.

S 1 y

 1 Þ Ñ ψpτ pyq, yq. (A.[START_REF] Motoya Machida | Λ-linked coupling for drifting Brownian motions[END_REF]

FIGURE 1 .and e θ 2 (

 12 FIGURE 1. The vectors e S 1 , N S pzq, e S 2 and e θ 2

  gpψpτ pyq, yqq det T ψpτ pyq, yq µpdyq `żD xdg, N y dµ.

S

  gpzq sin θpzq µpdzq `żD xdg, N y dµ. (A.23) APPENDIX B. MOVING SETS

  t`s rf spxq ´Pt rf spxq ´ż s 0 P t`u rLrf sspxq du Since Lrf s P R, the mapping r0, ss Q u Þ Ñ P t`u rLrf sspxq is right continuous, according to Lemma D.1, and the same argument as in (D.1) enables to conclude to (D.2).

tě0 ρpx 1

 1 ptq, x 2 ptqq `sup tě0 d β, F pD 1 ptq, D 2 ptqq, where for two domains D and D 1d β, F pD, D 1 q " " d β,D pD, D 1 q ^dβ,D 1 pD 1 , Dq ^ε if HpD, D 1 q ă ε ε otherwise. (G.2)

8 0, ż t 0 f n pX n s , D n s q ds ˙tě0 converges in law to ˆXt , D t , ż t 0 f

 800 f n : px, Dq Þ Ñ f n px, Dq and f : px, Dq Þ Ñ f px, Dq be maps on M ˆF with values in some Euclidean space, and U an open set in M ˆF for d 0 . Assume that: (i) the random variables ż |f n pX n s , D n s q| p ds are uniformly bounded in probability for some p ą 1, (ii) in the open set U , the functions f n converge locally uniformly to f with respect to d 0 , and are d 0 -continuous, (iii) for a.e. t ě 0, pX t , D t q P U . Then ˆXn t , D n t pX s , D s q ds ˙tě0 for pd α , | ¨|q.Remark G.2. In the applications we will always take(G.3) U " ! px, Dq P M ˆF , x P DzSpDq ) ,which is easily seen to be d 0 -open thanks to Assumption 3.1 on F .

(G. 8 )

 8 S :" pX s pωq, D s pωqq , s P rt ´ε1 , t `ε1 s (is d α -compact in M ˆF , so it has a d α -neighbourhood V included in U of the form (G.9)V "! px, Dq P M ˆF , d α ppx, Dq, Sq ď ε 2

  D pσ D q ˚pxq " Id TxM and ∆f D is bounded and uniformly Hölder continuous (due to Assumption 2.2). Notice that X t remains in D t , since when X t P BD t , we have, using (2.9) which yields on boundary xN Dt pX t q, σ Dt Dt pX t q dt ´xdBD t pX t q, N Dt pX t qy " xN Dt pX t q, dX t y ´1 2 h Dt pX t q dt ´xdBD t pX t q, N Dt pX t qy " 0. BDprq pxq " ρ BDp0q pxq ´r for x P Dp0q and |r| small, (see Appendix A). Dt pyq `bt ˙dt ˙, @y P BD t for some Brownian motion W t and some adapted locally bounded real-valued process b t .Let µ t " µ Dt be the Lebesgue measure on D t and μt " μDt " U pD t q " BDt the Lebesgue measure on BD t and μt " μBDt " Dt y ˘˘dt and dμ t pkq " p´μ t pkq `μ t pkqμ t pBD t qq dW t ´1 2 μt `xdk, N Dt y ˘dt `pμ t pBD t q `bt q p´μ t pkq `μ t pkqμ t pBD t qq dt Dt ´xdk, N Dt y ˘pyq µ t pdyq. q 3 d xµpD ¨qy t ´1 µpD t q 2 d xµ ¨pkq, µpD ¨qy t Dt yq dt `μ t pkqμ t pBD t qpdW t `bt dtq `μ t pkqμ t pBD t q 2 dt ´μ t pkqμ t pBD t q dt. Denote τ ε the exiting time of pD t q tě0 from F α,ε . As in Definition 2.1, we stop pX t , D t q tě0 at τ ε . " ∆f Dt pX t q we have dF k pD t q " ´µBDt pkq `dW t `∆f Dt pX t q dt and the covariation of gpX t q and F k pD t q is Γ L rg, F k spX t , D t q dt with (2.37) Γ L rg, F k spx, Dq " ´µBD pkqx∇g, ∇f D ypxq.

	Once we have a solution to (2.16), make by Girsanov theorem a change of probability In particular, This implies that
	such that pW t , W m t q is a Brownian motion where (2.19) W t :" x W t ´ż t 0 ˆµBDs pBD s q µpD s q (2.28) F 1 p0, kq " ´µpkq and F 2 p0, kq " µpkh ´xdk, N yq. `∆f Ds pX s q ˙ds. This allows us to compute (2.36) L pF k qpx, Dq " ´µBD pkq∆f D pxq ´1 2 µ BD `x∇k, N D y ˘,
	We get a solution to (2.10) in the new probability. (2.29) dpF pW t , kqq " F 1 pW t , kq dW t	`1 2	F 2 pW t , kq dt
	Proposition 2.9. Let D t satisfy and then, since dW t and xdBD t , N Dt yp¨q differ only by a finite variation process Consequently, using
	(2.20) function of M . Then dBD t pyq " N Dt pyq ˆdW t (2.32) dµpD t q " µ t pBD t q p´dW t ´bt dtq . `ˆ1 2 which gives (2.21). In particular, taking k " 1 we obtain by µ t " µ µ BDt µpD t q (2.31) dµ t pkq " ż BDt kpyq p´dW t ´bt dtq ´1 2 xdk, N Dt ypyq µ t pdyq dt, µpD t q . Let k be a smooth . Denote h µ Dt (2.30) dµ t pkq " ż BDt ´kpyqxdBD t pyq, N Dt pyqy `1 2 (2.38) L pgF k qpx, Dq " gpxqL pF k qpx, Dq `Fk pDq 1 2 ∆gpxq `ΓL rg, F k spx, Dq `kh This yields we get (2.34).
	The modified equation writes dBD t pyq " N Dt pyq ´dx W t `´1 2 h Dt pyq ´µBD t pBDtq µpDtq ¯dt ¯; dX t " ˜∇f Dt pX t q " d x W t ´´µ BD t pBDtq µpDtq `∆f Dt pX t q ¯dt ı for dµ t pkq " ´µt pkq dW t ´1 2 Now we can compute (2.16) $ ' ' ' ' ' ' ' & ' (2.21) dμ t pkq " d ˆµt pkq µpD t q " `2b t µ t pkq `µt `xdk, N (2.22) ' ' ' ' ' ' % `σDt c pX t q dW m t x W t and W m t independent Brownian motions. Notice that the first equation is the same as (2.11). Thus due to Theorem 2.5, pD t q tě0 is a diffusion process with generator Ă L . In particular, if b t " ´μ t pBD t q we get (2.23) dμ t pkq " p´μ t pkq `μ t pkqμ t pBD t qq dW t ´1 2 μt `xdk, N Dt y ˘dt. 1 µpD t q dµ t pkq ´µt pkq µpD t q 2 dµpD t q `µt pkq µpD " 1 µpD t q dµ t pkq ´µt pkq µpD t q 2 dµpD t q `µt pkq µpD t q 3 µpBD t q 2 dt ´1 µpD t q 2 µ t pkqµ t pBD t qdt
	Then given D t , the equation for X t dX t " ˜∇f Dt pX t q " d x W t Proof. Let us first work at fixed time t ě 0. Denote D " D t and adopt the corresponding ´ˆµ BDt pBD t q µpD t q `∆f Dt pX t q ˙dt  notations presented in Appendix A. For k a smooth function on M and r P R sufficiently close to 0 so that BDprq (defined in (A.3) and (A.4)) is a smooth manifold without bound-ary, let " ´μ t pkq pdW t `bt dtq ´1 2 μt pxdk, N This yields (2.22).
		(2.17)
		ż
	(2.24)	`σDt c pX t q dW m t F pr, kq "	k dµ.
		Dprq
	can also be solved, since the coefficients in front of d x W t and dW m t We have Proposition 2.10. Any solution of equation (2.10) stopped at τ ε is a Markov process solu-are Lipschitz, σ c pX t qdW m (2.25) F pr, kq " ż BD ˜ż τ pyq k pψpsqpyqq e ´şs tion to a martingale problem associated to a generator L acting in the following way: for 0 h D pψpuqpyqq du ds ¸µpdyq any g, k smooth functions on M and r ż t y " 0, with τ pyq the hitting time of SpDq by the inward normal flow started at y (defined in (A.1)) (2.33) F k pDq :" kdµ, dpρ BDt pX t qq " x∇ρ BDt , dX t y ´1 2 D and ψpsqpyq :" ψp0, sqpyq :" exp y psN y q defined in (A.5). The mapping h D is defined in (A.7) and is an extension of the mean curvature on the boundary BD: it corresponds to the we have for px, Dq P M ˆFα,ε , h (2.18) mean curvature for the foliation induced by the BDprq, r P R sufficiently small. With this (2.34)
	where we used (2.16) and (2.6). We also have no covariation since the martingale part of formulation we can differentiate with respect to r, to obtain (2.26) F 1 pr, kq " ´żBD kpψpr, yqqe ´şr L pgF k qpx, Dq " ´gpxq∆f D pxqµ BD pkq ´1 2 gpxqµ BD px∇k, N D yq F k pDq∆gpxq `1 2 0 h D pψpsqpyq ds µpdyq. ´µBD pkqx∇g, ∇f D ypxq.
	dBD t acts on the normal flow only, and any normal flow Differentiating again we get (2.27) F 2 pr, kq " ´żBD pxdk, B r ψpr, yqy ´pkhqpψpr, yqqq e Proof. From (2.10) and (2.21) with b t ˘´1 2 µ BDt `x∇k, N Dt y ˘dt. ´şr 0 h D pψpsqpyqq ds µpdyq. (2.35)

r Þ Ñ Dprq :" tx P M, ρ `pxq ě ru satisfies ρ t

  2 h Dt pyq ´hDt pX t q1 DtzSt pX t q ˙dt ´2 sinpθ St pX t qq dL St t pXq Here θ St pxq " π{2 ´ϕSt pxq, ϕ St pxq being the angle between the orthogonal line to S t at x and any of the two minimal geodesics from BD t to x P S t (recall S t is the regular skeleton of D t , see Appendix A). In other words θ St pxq is the smallest angle between S being the thickening of the regular part of S s in normal direction, of thickness β in both directions. Remark 3.6. Compared to Section 2 with f D replaced by distance to boundary ρ BD , we have outside the skeleton S D

	S β s (3.12)
		(3.10)
	(3.11)	L St t pXq " lim βOE0	1 2β	ż t 0	1 tXsPS β s u ds,

t and the geodesics. The process L St is the local time of X t at S t :" SpD t q:

  TxMϕ δ p|u|q pd i ρ δ pexp x puqq, BDq ´di ρ δ px, BDqq du(3.28)From this we see that d 1 f δ p¨, Dq converges, locally uniformly outside SpDq, to d 1 ρp¨, BDq with respect to the distance d 0 of Appendix G. We obtain, with Lemma G.

		1, possibly by
	again extracting a subsequence, that
	(3.29)
	.26)
	From Proposition 3.4, possibly by extracting a subsequence,
	(3.27)	`fδ pX δ t , D δ t q ˘tě0

L

ÝÑ pρpX t , BD t qq tě0 .

From (3.7) we get for i " 1, 2,

d i f δ px, Dq ´di ρ δ px, BDq

" ´di ε px, Dq ż TxM ϕ δ p|u|q pρ δ pexp x puqq, BDq ´ρδ px, BDqq du `p1 ´ ε px, Dqq ż

  x∇ 2 d 1 ρpX s , BD s q, dBD s b dX s y ρpX s , BD s q " `xN Ds pX s q, ¨y which implies that the covariant derivative in the second variable with respect to N Ds is equal to 0. On the other hand, by Itô-Tanaka formula (see Proposition E.1 in Appendix E using that ρpx, BDq is almost everywhere the minimum of two smooth functions) together with Assumption 3.1 which allows to only consider the regular skeleton, together with Theorem B.1 which says that the latter has absolutely continuous variation (useful for the term dL St Indeed, (3.25) implies that xd 1 ρpX t , BD t q, dX t y " dW t and due to (3.30), we have xd 2 ρpX t , BD t q, dBD t y

					˙tě0
				dX δ t yq
	(3.34)	ÝÑ L	ˆż t		˙tě0	" 0
				0
	since d 1 (3.35)
	Using (3.26), (3.27), (3.29), (3.30), (3.33), (3.34), (3.35) we obtain that
	(3.36)
	ˆż t		˙tě0
	∆ 1 f δ pX δ s , D δ s q ds
	0
	Remark 3.7. From (3.35), it can be deduced that
	(3.37)
		" lim δÑ0 xd 2 ρpX δ t , BD δ t q, dBD δ t y
		" lim δÑ0	´dW δ t ´ˆ∆ 1 f δ pP BD δ t pX δ t q, D δ t q	`1 2	h D δ t pP BD δ t pX δ t qq ˙dt
	where we used (3.21) in conjunction with (3.22).

t pXq), we have d pρpX t , BD t qq "xd 1 ρpX t , BD t q, dX t y ´1 2 h Dt pX t q1 DtzSt pX t q dt `xd 2 ρpX t , BD t q, dBD t y `0 `0 ´sin `θSt pX t q ˘dL St t pXq. L ÝÑ ˆż t 0 ´hDs pX s q1 DszSs pX s q ds ´ż t 0 2 sin `θSs pX s q ˘dL Ss s pXq ˙tě0 . It remains to pass in the limit as δ goes to zero in (3.21), to deduce (3.24). d pρpX t , BD t qq " 1 2 `hDt pX t q1 DtzSt pX t q ´hDt `P BDt pX t q ˘˘dt`sin `θSt pX t q ˘dL St t pXq. Taking into account (3.36), we identify the last limit with ´dW t `ˆh Dt pX t q1 DtzSt pX t q ´1 2 hpP BDt pX t qq ˙dt `2 sin `θSt pX t q ˘dL St t pXq 4. INTERTWINED DUAL PROCESSES: DECOUPLING AND REFLECTION ON BOUNDARY

  4.1) dBD t pyq " N Dt pyq ˆdW t `1 2 h Dt pyqdt ´dL BDt is a M -valued Brownian motion started at uniform law in D 0 , W t is a realvalued Brownian motion independent of X t , L BDt t pXq is the local time of X t on the moving boundary BD t .

	Remark 4.2. Equation (

t pXq ẇhere X t

  St pxq is naturally extended to D t by being constant on lines normal to the boundary (see[START_REF] Marc Arnaudon | Intertwining Brownian motions with symmetric convex sets[END_REF]). Notice that x Þ Ñ θ St pxq is locally Lipschitz on D t and is equal to 0 on D t X prx t , 8q ˆt0uq. Also notice that the function h Dt is locally Lipschitz on D t zS t . With these notations, equation (3.10) writes (again when ´π{n ă θ St pX t q ă π{n) dBD t pyq " ´N Dt pyq ˜signpX p1q t q cospθ St pX t qq dX Dt pyq ´hDt pX t q ˙dt ´2 sinpθ St pX t qqdL t pX p2q q

				.10) writes
	(5.31)	N Dt pX t q " ´signpX	p1q t q cospθ St pX t qqı ´signpX
	(5.32)
				p1q t	`sinpθ St pX t qqsignpX t q dX p2q t p2q
		`ˆ1 2	h ¸.

p2q

t q sinpθ St pX t qq where θ This equation written for ´π{n ă θ St pX t q ă π{n is enough to describe the whole coupling, thanks to the symmetry properties of D t .

Let us investigate the motion of the skeleton St of the solution Dt of equation (2.11) (garanteed by Theorem 2.5).

Proposition 5.5. The process ´D t ¯tě0

  N 1 pxqy ´xJ 1 p0q, N py 1 qy " xJ 2 p1q, N 2 pxqy ´xJ 2 p0q, N py 2 qy

					λ 2 sin θpxq pN 1 pxq ´N2 pxqq we
	have
	(B.11)	xJ 1 p1q, N 1 pxqy "	λ 2 sin θpxq	p1 ´cosp2θpxqq `xJ T 1 p1q, N 1 pxqy
		" λ sin θpxq `xJ T 1 p1q, N 1 pxqy
	and
	(B.12)	xJ 1 p1q, N 2 pxqy "	´λ 2 sin θpxq	p1 ´cosp2θpxqq `xJ T 1 p1q, N 2 pxqy
		" ´λ sin θpxq `xJ T 1 p1q, N 2 pxqy
	On the other hand we require that the variation of length of the two geodesics are the same.
	This writes as
	(B.13) xJ 1 p1q, or
	(B.14)	λ sin θpxq `xJ T 1 p1q, N 1 pxqy ´λ1 " ´λ sin θpxq `xJ T 1 p1q, N 2 pxqy ´λ2 ,
	which finally, with xJ T 1 p1q, N 1 pxq ´N2 pxqy " 0, yields λ "	λ 1 ´λ2 2 sin θpxq	, so the normal
	variation of S is given by
	(B.15)	xJ 1 p1q, N S 1 pxqyN S 1 pxq "	λ 1 ´λ2 2 sin θpxq	N S 1 pxq.

  Jp1, 0, ´ρS py 1 q∇H Dt py 1 qq. Again the processus z t does not play a role. To summarize, we have the following result for the evolution of S t : Theorem B.1. When D t evolves as (B.7) (B.26) dBD t pyq " N Dt pyqpH Dt pyq dt `dz t q,

	dt `pH Dt py 1 q ´HDt py 2 qqpN Dt `1 4 sin 2 θ St pxq 1 pxq `N Dt ˜2xJ K 1 p1q, N Dt 2 pxqyN Dt 1 pxq 2 pxqq ¸dt dS K t pxq " H Dt py 1 q ´HDt py 2 q 4 sin 2 θ St pxq ´N Dt 1 pxq ´N Dt 2 pxq ¯dt and the tangential evolution (B.25) which can be rewritten as (B.25) where J K (B.27) dS T t pxq 1 p1q " the regular skeleton S t has the normal evolution (B.23) " p S pJ K 1 p1qq dt `ˆ´x J K 1 pxqy 2 sin θ St pxq `HDt py 1 q ´HDt py 2 q 4 sin 2 θ St pxq ˙pN Dt 1 pxq `N Dt 2 pxqq dt 1 p1q, N S (B.28)

  where in the first equality we use the Itô formula, the fact that t Þ Ñ BG t is C 1`α 2 , Ψpx, rq " 0, and in the second equality we used Lemma 13 in[START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF], i.e. BD t is a solution in the Itô form :" dBD t pxq " pdW t `1 2 h BDt pxqdtqN BDt pxq x P BD t . (C.5) Proposition C.2. Conversely, if BD t is a solution of (C.5) then BG t " ΨpBD t , W t q is a solution of (C.2). Proof. Let x P BΨpBD t , W t q dΨpBD t , W t qpxq " T 1 Ψ pBDt,Wtq p˝dBD t qpxq `νΨpBDt,Wtq pxqdW t " T 1 Ψ pBDt,Wtq ppdW t `1 2 h BDt dtqN BDt qpxq ´N ΨpBDt,Wtq pxqdW t BDt pΨ ´1pBD t , W t qpxqqN BGt pxqdt ΨpBGt,´Wtq pΨpBG t , ´Wt qpxqqN BGt pxqdt (C.6) where we use that in this case, the Stratonovich differential is equal to the Itô's one (c.f. Appendix B), i.e. ˝dBD t pxq " dBD t , and d 2 d 2 r Ψpx, rq " 0. So BG t is a solution of (C.2).

				´Wt qpxq "
	(C.4)	" T 1 Ψ pBGt,´Wtq p ´νΨpBGt,Wtq pxqdW t d BG t qpΨ ´1pBG t , ´Wt qpxq dt dt
		" ˆdW t	`1 2	h ΨpBGt,´Wtq pxqdt ˙N ΨpBGt,´Wtq pxq,
	d 2
	d 2 r "	ˆ1 2	h "
			1 2	h

  pX t qs ´Erf pX 0 qs ´E "ż t

				
				gpX s q ds	" 0
				0
	Using Fubini's lemma (applicable due to our measurability requirement on Ω) and tak-
	ing into account the definition of P , we get
		P t rf spxq ´P0 rf spxq	´ż t	P s rgspxq ds " 0
				0
	namely, recalling that we required that g P R,
		g " P 0 rgs
			ż t
		" lim tÑ0`1 t	0	P s rgspxq ds
	(D.1)	" lim tÑ0`P t rf spxq ´f pxq t

  spX s 1 ´sqs ´Ey rP t rf spX 0 qs ´ż s 1 ´s

0 E y rP t rLrf sspX u qs du " P t`s 1 ´srf spyq ´Pt rf spyq ´ż s 1 ´s 0 P t`u rLrf sspyq du Taking into account (D.2), the last integral is equal to ż s 1 ´s 0 B u P t`u rf spyq du " P t`s 1 ´srf spyq ´Pt rf spyq which ends the proof of (D.3).

  2 ∆pf `gqpX t qdt `x∇pf `gqpX t q, dX t y 1 2 ´signppf ´gqpX t qqdppf ´gqpX t qq `dL 0,t ppf ´gqpX . qq ¯,where L 0,t ppf ´gqpX . qq " lim εÑ0 `1 ε ş t 0 1 r0,εs ppf ´gqpX s qqdxpf ´gqpXq, pf ´gqpXqy s . Since locally S " tf ´g " 0u and µpSq " 0, we have dρ BD pX t q " 1 2 1 XtRS ∆ρ BD pX t qdt `1XtRS x∇ρ BD pX t q, dX t y ´1 2 dL 0,t ppf ´gqpX . qq. XtRS ∆ρ BD pX t qdt `1XtRS x∇ρ BD pX t q, dX t y r´ε,εs ppf ´gqpX s qq}∇pf ´gq} 2 pX s q ds.In Appendix A it is shown that for x P S , }∇pf ´gqpxq} " 2 sin `θS pxq ˘.

	After changing the role of f and g we get
	(E.1)
	dρ BD pX t q "	1 2	1 ´1 2	dL 0 t ppf ´gqpX . qq,
	where
	L 0 t ppf ´gqpX . qq " lim εÑ0 `ż t 0 1 Using the flow d 1 2ε dt γptq " ´∇pf´gqpγptqq

}∇pf ´gqpγptqq} 2 that starts at y P U , we get ty P M, s.t. |f ´g|pyq ď εu Ă ty P M, s.t. |d S pyq| ď ε 2 sin pθ S pγpgpyqqqq `opεqu, where d S is the distance to S. On the other hand, using the minimal geodesic from S to y P U we get ty P M, s.t. |d S pyq| ď εu Ă ty P M, s.t. |f ´g|pyq ď 2ε sin `θS pP S pyqq ˘`opεqu.

  equation (F.3) becomes in a differential form (F.4) dF k pD t q ´x L rF k spD t qdt " p

				ż BD	k dσq `dB t	`µBDt pBD t q µpD t q	dt ˘.
	Let
	M t " e	´şt 0 x	µ BDs pBDs q µpDsq	, dBsy´1 2	ş t 0	`µBDs pBDs q µpDsq	˘2 ds ,
			P

|Ft " M t Q|Ft .

  Then sup sPrt´ε,t`εs |f n pX n s pωq, D n s pωqq ´f pX s pωq, D s pωqq|

	ď	sup	|f n pX n s pωq, D n s pωqq ´f pX n s pωq, D n s pωqq|
		sPrt´ε,t`εs
		`sup

sPrt´ε,t`εs |f pX n s pωq, D n s pωqq ´f pX s pωq, D s pωqq| .

  Consequently, using (G.12) and (G.16), we get ω a.s. for a.e. t ě 0 (G.17) t pωq " f pX t pωq, D t pωqq Integrating we get ω-a.s. for all t ě 0 (G.18) a t pωq " A t pωq " ż t 0 f pX s pωq, D s pωqq ds.

	On the other hand
	ˇˇˇ1 2ε	ż t`ε t´ε	s pωq ´ t pωq ds ˇˇˇď 1 2ε	ż t`ε t´ε	| s pωq ´ t pωq| ds
	so (G.14) implies that ω a.s. for a.e. t ě 0
	(G.16)		lim εOE0	1 2ε	ż t`ε t´ε	s pωq ds " t pωq.
						ż t
	13)				a t pωq "	s pωq ds.
						0
	By Lebesgue theorem, ω a.s., for a.e. t ě 0
	(G.14)		εOE0 lim	2ε 1	ż t`ε

t´ε | s pωq ´ t pωq| ds " 0. Equalities (G.12) and (G.13) imply that ω a.s. (G.15) lim εOE0 1 2ε ż t`ε t´ε s pωq ds " f pX t pωq, D t pωqq for a.e. t.

Funding from the grant ANR-17-EURE-0010 is aknowledged by L.M.