Koléhè Coulibaly-Pasquier

Laurent Miclo

On the evolution by duality of domains on manifolds Sur l'évolution par dualité de domaines dans des variétés

Keywords: Elliptic diffusions, Riemannian manifolds, stochastic domain dynamics, duality by intertwining, Bessel processes, modified mean curvature flows, constant curvature spaces Diffusions elliptiques, variétés riemanniennes, dynamiques stochastiques de domaines, dualité par entrelacement, processus de Bessel, flots par courbure moyenne modifiés, espaces à courbure constante primary: 60J60, secondary: 58J65, 53C21, 53C44, 60J35, 35K59

On a manifold, consider an elliptic diffusion X admitting an invariant measure µ. The goal of this paper is to introduce and investigate the first properties of stochastic domain evolutions pD t q tPr0,τs which are intertwining dual processes for X (where τ is an appropriate positive stopping time before the potential emergence of singularities). They provide an extension of Pitman's theorem, as it turns out that pµpD t qq tPr0,τs is a Bessel-3 process, up to a natural time-change. When X is a Brownian motion on a Riemannian manifold, the dual domain-valued process is a stochastic modification of the mean curvature flow to which is added an isoperimetric ratio drift to prevent it from collapsing into singletons.

Résumé

Sur une variété, considérons une diffusion elliptique X de mesure invariante µ. Le but de ce papier est d'introduire et d'étudier les premières propriétés d'évolutions stochastiques de domaines pD t q tPr0,τs qui sont des processus duaux par entrelacement de X (où τ est un temps d'arrêt strictement positif précédant l'apparition éventuelle de singularités). Il s'agit d'une extension du théorème de Pitman, puisqu'il ressort que pµpD t qq tPr0,τs est un processus de Bessel-3, à un changement naturel de temps près. Quand X est un mouvement brownien sur une variété compacte, ce processus dual à valeurs domaines est une modification stochastique du flot par courbure moyenne auquel est ajouté une dérive fournie par un quotient isopérimétrique qui l'empêche de s'effondrer en des singletons.

Introduction

In the finite state space framework, Diaconis and Fill [START_REF] Diaconis | Strong stationary times via a new form of duality[END_REF] have shown that ergodic Markov chains can be intertwined with Markov chains living on the set of non-empty subsets of the state space and ending up being absorbed at the full state space. This result enabled them to construct strong stationary times for ergodic Markov chains, leading to quantitative bounds on their convergence to equilibrium, in the separation discrepancy and in the total variation distance. In [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF], this point of view was extended to real ergodic diffusion processes, but the one-dimensionality seemed crucial in the method. As noted in this previous paper, it is quite unfortunate, since otherwise it could lead to a new probabilistic approach to the hypoellipticity theorem of Hörmander [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF]. Here we make an important step further in this program, by showing that elliptic diffusions on differential manifolds admitting an invariant measure can indeed be intertwined with domain-valued Markov processes. Although the hypoellipticity is not yet entering in the game (but see [START_REF] Miclo | Duality and hypoellipticity: one-dimensional case studies[END_REF] for a first illustration in dimension 1), the introduced domain-valued processes are already very intriguing and promising for themselves. When dealing with the Brownian motion on a Riemannian manifold, they are natural stochastic modifications of the mean curvature flow. In the more general case, when a gradient drift is added to the Brownian motion, one has to consider some weighted extensions.

Let L be an elliptic diffusion generator on a differentiable manifold V . Here we will not be interested in regularity problems, so V and the coefficients of L are supposed to be smooth. Assume there exists a σ-finite measure µ on V which is invariant for L in the sense that @ f P C 8 c pV q, µrLrf ss " 0 where C 8 c pV q stands for the space of smooth functions on V with compact support. By ellipticity, the measure µ admits a positive density with respect to the Riemannian measure. Note that in general µ is not unique, even up to a positive factor, e.g. for the generator B 2 `B on R, all the measures with a density of the form R Q x Þ Ñ a `b exppxq, with a, b ě 0, are invariant. But there is at most one finite invariant measure and in this case it is usual to normalize µ into a probability measure.

Let D be the set of non-empty, compact and connected domains D in V , which coincide with the closure of their interior and whose boundary C BD is smooth. Denote also D D \ ttxu : x P V u, obtained by adjunction of all the singletons to D, and D the set of all measurable subsets D of V which either satisfy µpV q P p0, `8q or are singletons (so that D Ă D Ă D). Consider the Markov kernel Λ from D to V given by @ D P D, @ A P BpV q, ΛpD, Aq

$ & % µpA X Dq µpDq , if µpDq ą 0 δ x pAq , if D " txu, with x P V (1)
where BpV q is the set of measurable subsets of V and δ x the Dirac mass at x. As usual, such an integral kernel can be seen as an operator transforming bounded (respectively positive) measurable functions on V into finite-valued (resp. p0, `8s-valued) functions on D.

The main goal of this paper is to find a Markov generator L with state space D satisfying, in an appropriate sense, the intertwining relation

LΛ " ΛL (2)
and for which the singletons are entrance boundaries.

Remark 1 This was done in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF] when V " R and when ´8 and `8 were entrance boundaries for L. The latter assumption was needed to insure that the resulting Markov processes on the set of the closed segments (which were not assumed to be compact in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF]) end up being absorbed at the whole state space R, because we were primarily interested in constructing strong stationary times. This is no longer our objective here (even if we should come back to this question in a future work), that is why no assumption is made on the behavior of L at infinity. Note also that in general there is not a unique Markov generator satisfying the above requirements, since in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF] we constructed a whole family of such operators when V " R. Nevertheless, among them, one was the fastest to be absorbed at R, it is a generalization of this Markov generator that will be considered below.

As a consequence of the previous remark, from now on, we assume that the dimension of V is larger or equal to 2. To describe our candidate L, we need to introduce some notations.

By using the inverse of the matrix diffusion of L to induce a Riemannian structure on V (see e.g. the book [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] of Ikeda and Watanabe for the details), L can be decomposed as L " `b, where is the Laplacian operator associated to the Riemannian structure and b is a vector field (seen as a first order differential operator). We assume that V is complete, endowed with the Riemannian distance d. Let λ be the Riemannian measure on V . It is well-known that µ is absolutely continuous with respect to λ and that its density is smooth. Let us write U lnpdµ{dλq P C 8 pV q (a priori defined up to an additive constant, except when µ is normalized into a probability measure). The vector field b can written as ∇U `β, with the vector field β satisfying divpexppU qβq " 0; it corresponds to the µ-weighted Hodge decomposition of b. In the previous sentence, ∇ and divp¨q are the gradient and divergence operators associated to the Riemannian structure. Other Riemannian notions that will be useful for our purpose are the scalar product x¨, ¨y, as well as the exterior normal vector ν C , the "mean" curvature ρ C and the pdimpV q ´1q-Hausdorff measure σ C , all the last three objects being defined on the boundary C of an element D P D. The mean curvature is signed with respect to our choice of the orientation of ν C and it is not really a mean, since it is the trace (without renormalization) of the second fundamental form. A priori the orientation of ν C and the sign of ρ C require to know on which side of C is the interior of D (except when V is not compact, then the mapping D Q D Þ Ñ C is one-to-one, otherwise it is two-to-one), but ρ C ν C depends only on C.

Let us first describe heuristically the type of stochastic evolution pD t q tPr0,τq in D we want to consider. The positive stopping time τ is earlier than the exit time from D, typically due to the apparition of singularities on the boundary C t BD t . We want, as long as t P r0, τq, the infinitesimal evolution of any Y t P C t to be given by dY t " ˆ?2dB t `ˆ2 σ Ct pexppU qq µpD t q `xβ ´∇U, ν Ct y pY t q ´ρCt pY t q ˙dt ˙νCt pY t q

where B pB t q tě0 is a standard real Brownian motion. The evolution (3) can be seen as a deterministic and stochastic modification of the mean curvature flow, which corresponds to dy t " ´ρCt py t qν Ct py t q dt for the points y t on the evolving boundary. The global term σ Ct pexppU qq{µpD t q (it does not depend on the position of Y t on C t ) in (3) can be seen as an isoperimetric ratio with respect to µ. Indeed, it can be rewritten as µpC t q{µpD t q, where µ is the pdimpV q ´1q-dimensional measure on C t admitting exppU q as density with respect to σ Ct . So this term explodes as D t becomes closer and closer to a point. In some sense, it will compensate the trend of the mean curvature flow on compact boundaries to make them smaller and smaller (and rounder and rounder). Though too qualitative to be convincing, this observation is a first hint of why the singletons will be entrance boundaries for the Markov processes determined by [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF].

The term xβ, ν Ct y pY t qν Ct pY t q in (3) could be replaced by βpY t q, since the tangential components in the description of the evolution of the points on boundary can be removed, up to a diffeomorphism of C t (see e.g. Section 1.3 of Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF]). Only the radial component (i.e. the projection on the normal vectors ν Ct ) is important, thus an equation such as [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] will be said to be radial.

In fact, the radial stochastic differential equation (3) of the points on the boundary is not the most convenient way to work with the process pD t q tPr0,τq . In Markov theory, the martingale problem approach is usually more helpful (for a general introduction and an extensive development of this notion, cf. for instance the book of Ethier and Kurtz [START_REF] Stewart | Markov processes[END_REF]). It needs convenient observables on the state space. On D, the role of elementary observables is played by the mappings

F f : D Q D Þ Ñ F f pDq ż D f dµ (4)
associated to the functions f P C 8 pV q, the space of smooth mappings on V .

To proceed in the direction of the definition of the generator L on an appropriate algebra D of functionals defined on D, we begin by defining the action of L on the above elementary observables: for any f P C 8 pV q, @ D P D, LrF f spDq

ż D Lrf s dµ `2 µpCq µpDq ż C f dµ (5)
Using Stokes formula, we will check in Section 3 that the above r.h.s. can be written as an integral over C only:

@ D P D, LrF f spDq " ż C x∇f, ν C y `ˆ2 µpCq µpDq `xβ, ν C y ˙f dµ (6)
Furthermore, we introduce a bilinear form Γ L (which will be the carré du champs associated to L) on such functionals, via @ f, g P C 8 pV q, @ D P D,

Γ L rF f , F g spDq ˆżC f dµ ˙ˆż C g dµ ˙(7)
Since the D-valued Markov processes we are interested in will have continuous sample paths (namely they will be diffusions), we are naturally led to the following definitions (see e.g. the book of Bakry, Gentil and Ledoux [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]). Consider D the algebra consisting of the functionals of the form F fpF f 1 , ..., F fn q, where n P Z `, f 1 , ..., f n P C 8 pV q and f : R Ñ R is a C 8 mapping, with R an open subset of R n containing the image of D by pF f 1 , ..., F fn q. For such a functional F, define LrFs

ÿ jP 1,n B j fpF f 1 , ..., F fn qLrF f j s `ÿ k,lP 1,n B k,l fpF f 1 , ..., F fn qΓ L rF f k , F f l s (8)
To two elements of D, F fpF f 1 , ..., F fn q and G gpF g 1 , ..., F gm q, we also associate Γ L rF, Gs ÿ lP n ,kP m B l fpF f 1 , ..., F fn qB k gpF g 1 , ..., F gm qΓ L rF f l , F g k s (9)

Remark 2 A priori the above definitions are ambiguous, since they seem to depend on the writing of F P D under the form fpF f 1 , ..., F fn q and similarly for G. To see that they are indeed well-defined, note that @ F, G P D, Γ L rF, Gs " 1 2 pLrFGs ´FLrGs ´GLrFsq

This property implies that if f is a polynomial in n variables, then for any F fpf 1 , ..., f n q, with f 1 , ..., f n P C 8 pV q, the object LrFs is uniquely defined. Indeed, it relies on an iteration on the degree of f, starting from ( 6) and [START_REF] Stewart | Markov processes[END_REF]. The general case of smooth functions f is deduced from their approximation over compact domains by polynomial mappings.

Let us come back to the Markov operator Λ defined in [START_REF] Anderson | Hyperbolic geometry[END_REF]. For any f P C 8 pR 2 q, Λrf s is an element of D, since it can be written

@ D P D, Λrf spDq " F f F 1 pDq
where 1 is the constant function taking the value 1. This relation also leads us to endow D with the σ-algebra generated by the mappings F f , for f P C 8 pV q, so that Λ is really a Markov kernel from D to V : for any fixed A P BpV q, the mapping D Q D Þ Ñ ΛpD, Aq is measurable. For this mapping to be measurable on D, put on the set tδ x : x P V u the σ-algebra obtained by identifying it with V (seeing δ x as x) and consider on D the σ-algebra generated by those on D and on tδ x : x P V u. Since we already mentioned continuity of trajectories, we must also endow D with a topology. The simplest way to do so would be to consider the smallest topology such that all the mappings F f , for f P C 8 pV q, are continuous (with the natural extension that the F f vanish on the singletons). But for our purpose, we will need a stronger topology making continuous the following functionals, for any f P C 8 pV q:

D Q D Þ Ñ ΛrF f spDq (10)
D Q D Þ Ñ ż C f dµ (11)
with the convention that if D is a singleton, then C " H (so that the latter r.h.s. is 0). Condition [START_REF] Gallot | Riemannian geometry. Universitext[END_REF] enables us to topologically identify tδ x : x P V u with V . The topology on D will be such that the σ-algebra put on D is the Borelian one. Condition [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF] implies that for any f P C 8 pV q, LrF f s is continuous on D. For the precise definition of this topology, see Section 3, where D will furthermore be endowed with an infinite-dimensional differential structure. After these structural precisions, let us come back to L, whose main interest is to fulfill our goal (2):

Theorem 3 For any f P C 8 pV q, we have

@ D P D, LrΛrf sspDq " ΛrLrf sspDq
To go further, we want to construct Markov processes whose generator is L and to establish a link with [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF].

Let be given a filtered probability space pΩ, F, pF t q tě0 , Pq, all subsequent notions from stochastic process theory will be relative to the filtration pF t q tě0 . Consider a stopped continuous and adapted stochastic process pD t q tPr0,τq , taking values in D and where τ is a positive stopping time. It is said to be a solution to the martingale problem associated to pD, Lq, if for all t P p0, τq, D t P D and if for any F P D, the process M F pM F t q tPr0,τq defined by @ t P r0, τq, M F t FpD t q ´FpD 0 q ´ż t 0 LrFspD s q ds is a local martingale. More precisely, in this situation we say that pD t q tPr0,τq is a solution to the martingale problem associated to the generator pD, Lq and to the initial distribution LpD 0 q, the law of D 0 , or starting from D 0 P D, when LpD 0 q is a Dirac mass.

One key to the following result is the adaptation of the Doss [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF] and Sussman [START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF] method to the infinite dimensional stochastic differential equation [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF].

Theorem 4 For any D 0 P D, there is a solution to the martingale problem associated to pD, Lq starting from D 0 .

In certain homogeneous spaces, it is possible to start from singletons, because these situations can be brought back to the 1-dimensional setting treated in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF]. Indeed, the processes pD t q tě0 end up being balls centered at the point from the initial singleton and it is sufficient to study the evolution of the radius. This is the case of the Laplacian operator on Euclidean, hyperbolic and spheric spaces. The stopping time τ is infinite in the two former situations and corresponds to the hitting time of the whole sphere in the latter one. But in general to consider D as state space is probably too restrictive. We believe there exists a set G of subdomains of V , with D Ă G Ă D, such that L can be naturally extended to G, in particular one should be able to define µ and ν BD , µ-a.e. Heuristically, the set of singular points of the boundary of a domain from G should be very small. We hope to investigate this question in a future work via the geometric measure theory, but for the moment being, let us assume that we are given such a set G with Theorem 4 holding up to a positive stopping time earlier than the exit time of G. Still denote by pD t q tPr0,τq the corresponding Markov processes. Consider ς 2 ż τ 0 pµpC s qq 2 ds P p0, `8s

and the time change pθ t q tPr0,ςs defined by @ t P r0, ςs, 2

ż θt 0 pµpC s qq 2 ds " t. (13)
Theorem 5 The process pµpD θt^ς qq tě0 is a (possibly stopped) Bessel process of dimension 3.

By taking into account that 0 is an entrance boundary for the Bessel process of dimension 3, a consequence of Theorem 5 is that the set of singletons is an entrance boundary for the Markov processes associated to pD, Lq, if we were able to extend Theorem 4 to initial conditions that are singletons. Theorem 5 can be seen as a multidimensional extension of the intertwining relation between the real standard Brownian motion and the Bessel process of dimension 3 by Pitman [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF]: it corresponds to [START_REF] Arnaudon | Construction of set-valued dual processes on manifolds[END_REF] when L is the Laplacian on R (see also Remark 37 in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF]).

Up to now, we did not consider the Markov processes associated to L, whereas their study is the first motivation for the above developments. The martingale problems associated to pC 8 pV q, Lq are well-posed (see e.g. the book of Ikeda and Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]), so to any initial distribution on V , we can associate a stopped Markov process pX t q tPr0,τ q where τ is the explosion time (maybe infinite). The conjunction of Theorems 3 and 4 should lead to the following result, which is the reason behind our interest in the relation (2): Conjecture 6 Assume that the martingale problems associated to pC 8 pV q, Lq are well-posed and defined for all times (no explosion). Let x 0 P V be given and let X pX t q tě0 be a solution starting from x 0 P V for the martingale problems associated to pC 8 pV q, Lq. Up to enlarging the underlying probability space, it is possible to couple the trajectory pX t q tě0 with a solution pD t q tPr0,τs starting from the singleton tx 0 u to the martingale problem associated to pD, Lq, such that for any stopping time T with T ď τ, we have for the conditional laws:

LpD r0,T s |Xq " LpD r0,T s |X r0,T s q (14) LpX T |D r0,T s q " ΛpD T , ¨q (15)
The difficulty behind the proof of such a result is technical, since conceptually it is an immediate extension of the ideas of Diaconis and Fill [START_REF] Diaconis | Strong stationary times via a new form of duality[END_REF] in the context of finite Markov chains. Two different approaches to such couplings for diffusions via coalescing stochastic flows have been proposed in Machida [START_REF] Machida | Λ-linked coupling for drifting Brownian motions[END_REF] and [START_REF] Miclo | On the construction of set-valued dual processes[END_REF], but they would need to be developed further to deal with the generality of our present framework. A related point of view is currently under construction in [START_REF] Arnaudon | Construction of set-valued dual processes on manifolds[END_REF]. Note that Conjecture 6 would enable us to come back to our initial motivation, first by recovering the density theorem for elliptic diffusions:

Corollary 7 Assume that a coupling of pX t q tě0 with pD t q tPr0,τs can be constructed as in Conjecture 6. Then for any t ą 0, the restriction to V of the law of X t is absolutely continuous with respect to the Riemannian measure λ.

To obtain this result, only the existence of a domain-valued dual process is needed (as well as its coupling with the process X), its uniqueness is irrelevant. The well-posedness of the martingale problems associated to pD, Lq is not crucial for this kind of consideration, more important for us would be the possibility for the dual process to start from singletons.

Another interesting stochastic domain evolution is obtained by removing the isoperimetric ratio from the generator, namely corresponding to the generator pD, r Lq, where ( 5) is replaced by

@ D P D, r LrF f spDq ż D Lrf s dµ (16)
for its action on the elementary observables (but ( 7), ( 8) and ( 9) remain unchanged). The associated Markov processes are the analogues of the evolving sets considered by Morris and Peres [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF] in discrete settings. One downside of the processes p r D t q tPr0,τq associated to the generator pD, r Lq is that they have a strong tendency to collapse in singletons in finite time and they remain singletons when starting from a singleton. The heuristic reason behind this collapse is that pµp r D t qq tPr0,τq is a non-negative martingale, due to r LrF 1 s " 0. Thus, assuming for instance that τ " `8, pµp r D t qq tPr0,`8q must converge in large time, as well as its bracket. It follows that lim inf tÑ`8 µp r C t q " 0 and appropriate geometric assumptions will enable to conclude that r D t becomes closer and closer to a singleton, at least along a sequence of diverging times (in the same spirit, an isoperimetric-type inequality between µ and µ will imply that lim tÑ`8 µp r D t q " 0). The convergence toward a singleton can be checked rigorously when starting from a ball in the constant curvature framework of the next section. In fact, taking into account the general theory of Doob transforms (with respect to the mappings D Q D Þ Ñ µpDq), the processes pD t q tPr0,τq correspond to the process p r D t q tPr0,τq conditioned not to hit the set of singletons, or more precisely, conditioned so that pµp r D t qq tPr0,τq does not hit zero. This property gives an understanding of the emergence of the Bessel-3 process in Theorem 5, seen as the Brownian motion conditioned not to hit 0 (see also the observations at the end of Section 7).

The plan of the paper is as follows. In the next section, we will deal with the simple but illustrative situation of the Euclidean, spheric and hyperbolic Brownian motion starting from a point. In Section 3 we prove Theorem 3 and Theorem 4. Section 4 presents a result on the existence of stochastic modified mean curvature flows, which was required by the proof of Theorem 4. Section 5 comes back to the homogeneous situations of Section 2, pursuing further some computations relative to the mean curvature addressed in Section 3. It will also show some critical differences between two ways of applying the Doss-Sussman method in these homogeneous geometric frameworks. In Section 6, Theorem 5 is proved as well as other properties of the solutions to the martingale problems associated to pD, Lq.

In particular, we will see that if the evolution pD t q tě0 is defined for all times, relatively to the usual Laplacian L " on the plane, then renormalizing the domains so that their areas is brought back to 1, we get a convergence in large time toward the disk centered at 0 of radius 1{ ?

π. An appendix provides supplementary informations on product situations and alternative dual processes (on domains whose boundaries are naturally non smooth).

Homogeneous situations

There are examples where the radial evolution equation ( 3) can be globally solved by coming back to the one-dimensional situation as it is treated in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF] (see also Fill and Lyzinski [START_REF] Allen | Strong stationary duality for diffusion processes[END_REF]). They correspond to spaces V with constant curvature endowed with the Laplacian and we take µ " λ and µ " σ C (denoted σ, to simplify), for C " BD and D P D, with the notation of the introduction. For them, we investigate solutions pD t q tě0 of the form pBp0, R t qq tě0 , where 0 is any fixed point of the state space, Bp0, rq is the closed ball centered at 0 of radius r ě 0 and pR t q tě0 is a R `-valued diffusion process starting from 0. We will describe separately the three situations of null, negative and positive constant curvature spaces.

Euclidean spaces

We consider here the Euclidean space R n , with n P Nzt1u. Without loss of generality, we can assume that 0 is the point zero from R n . For r ą 0, the Lebesgue volume of Bp0, rq is λpBp0, rqq " π n{2

Γpn{2`1q r n and the corresponding hypersurface volume of the sphere BBp0, rq is σpBBp0, rqq " n π n{2

Γpn{2`1q r n´1 . The mean curvature of any element x P BBp0, rq is ρpxq " pn ´1q{r. Thus a solution pBp0, R t qq tě0 of the radial evolution equation ( 3), is given by

dR t " ? 2dB t `ˆ2n R t ´n ´1 R t ˙dt " ? 2dB t `n `1 R t dt
where pB t q tě0 is a standard Brownian motion. Thus pR t{2 q tě0 has for generator the operator A given by

@ f P C 8 pR `q, @ x P R `, Arf spxq 1 2 f 2 pxq `n `1 2x f 1 pxq
(in the sequel such a generator will be denoted 1 2 B 2 `n`1 2x B), namely it is a Bessel process of dimension n `2. In particular 0 is an entrance boundary for pR t q tě0 and we can make it start from 0, i.e. we can let pBp0, R t qq tě0 start from t0u.

Let us check directly that pλpBp0, R t qqq tě0 is a Bessel of process of dimension 3, up to a time change, as announced in Theorem 5. It is sufficient to show that the same is true for pR n t q tě0 . We compute

dR n t " nR n´1 t ˆ?2dB t `n `1 R t dt ˙`2 npn ´1q 2 R n´2 t dt " ? 2nR n´1 t dB t `2n 2 R n´2 t dt
So the generator of pR n t q tě0 is 2n 2 x 2´2{n r 1 2 B 2 `1 x Bs. It follows that pR n θt q tě0 is a Bessel process of dimension 3, where the time change pθ t q tě0 is defined by

@ t ě 0, ż θt 0 R 2´2n s ds " 2n 2 t

Spherical spaces

We consider now the sphere S n Ă R n`1 , with n P N. Without loss of generality, we can assume that 0 is the point p1, 0, 0, .., 0q from R n`1 . For any r P r0, πs, Bp0, rq is the closed cap centered at 0 of radius r. In particular, we have Bp0, 0q " t0u and Bp0, πq " S n . Let λ be the uniform distribution on S n and σ be the corresponding hypersurface volume. The projection of λ on the first coordinate of R n`1 is the measure Z ´1 n p1 ´x2 q n{2´1 1 r´1,1s pxq dx, where the renormalising factor is given by the Wallis integral

Z n " ż 1 ´1p1 ´x2 q n{2´1 dx " ż π 0 sin n´1 puq du " ? π Γ `n`1 2 Γ `n 2 `1T
he cap Bp0, rq is exactly the set of elements of S n whose first coordinate belongs to rcosprq, 1s. So we get

λpBp0, rqq " Z ´1 n Iprq Z ´1 n ż r 0 sin n´1 puq du σpBBp0, rqq " Z ´1 n sin n´1 prq
The mean curvature of any element x P BBp0, rq is ρpxq " pn ´1q cotprq. Indeed, the mean curvature ρ on BBp0, rq is the function such that for any C 8 pS n q, we have

B r ż BBp0,rq f dσ " ż BBp0,rq
x∇f, νy dσ `żBBp0,rq f ρ dσ (for more details, see e.g. Lemma 10 in Section 3 below). Due to the symmetries of BBp0, rq, one sees that ρ must be constant on BBp0, rq. Thus considering f " 1 in the above equality, we get ρ " B r σpBBp0, rqq σpBBp0, rqq " pn ´1q cotprq It follows that a solution pBp0, R t qq tPr0,τq of the radial evolution equation [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF], where τ is the hitting time of π by pR t q tPr0,τq , is given by dR t " ? 2dB t `ˆ2 sin n´1 pR t q IpR t q ´pn ´1q cotpR t q ˙dt (17

)
where pB t q tě0 is a standard Brownian motion. As r Ñ 0 `, we have 2 sin n´1 prq IpR t q ´pn ´1q cotprq "

2r n´1 ş r 0 u n´1 du ´n ´1 r " n `1 r
and this enables us to see that 0 is an entrance boundary for pR t q tPr0,τq and we can make it start from 0, namely we can let pBp0, R t qq tPr0,τq start from t0u.

In general we did not find a nice expression for the drift of ( 17), but in the case n " 2, this evolution equation can be written dR t " ? 2dB t `2 `cospR t q sinpR t q dt

Similarly to the Euclidean situation, let us check directly Theorem 5, i.e. that pλpBp0, R t qqq tPr0,τq is a stopped Bessel of process of dimension 3, up to a time change. It is sufficient to show that the same is true for pIpR t qq tPr0,τq . We compute dIpR t q " I 1 pR t q ˆ?2dB t `ˆ2 sin n´1 pR t q IpR t q ´pn ´1q cotpR t q ˙dt ˙`I 2 pR t qdt " ? 2 sin n´1 pR t qdB t `ˆ2 sin 2n´2 pR t q IpR t q ´pn ´1q sin n´1 pR t q cotpR t q ˙dt `pn ´1q sin n´2 pR t q cospR t qdt " ? 2 sin n´1 pR t qdB t `2 sin 2n´2 pR t q IpR t q dt So the generator of pIpR t qq tPr0,τq is 2 sin 2n´2 pI ´1pxqqr 1 2 B 2 `1 x Bs, where I ´1 is the inverse mapping of I : r0, πs Ñ r0, Z n s. This shows that pIpR θt qq tPr0,τq is a Bessel process of dimension 3 starting from 0 and stopped when it hits Z n , where the time change pθ t q tPr0,τq is defined by

@ t P r0, τq, ż θt 0 1 sin 2n´2 pI ´1pR s qq ds " 2t
Consider the case where R 0 " 0. Then θ τ has the same law as the first hitting time of Z n by a Bessel process of dimension 3 starting from 0. It follows that τ is a.s. finite. Thus, starting from t0u, the process pBp0, R t qq tPr0,τs ends up covering the whole sphere S n at the (a.s.) finite time τ. According to the theory of strong duality (see e.g. the initial paper of Diaconis and Fill [START_REF] Diaconis | Strong stationary times via a new form of duality[END_REF] for the principe and Section 7 for its application to the present context), this property leads to the construction of strong stationary times for the Brownian motion on S n starting from 0 (and more generally for any initial distribution on S n , by symmetry and conditioning with respect to the initial position of the spheric Brownian motion).

Hyperbolic spaces

Consider the Poincaré's ball model of the hyperbolic space H n of dimension n P Nzt1u. For references on the subject, one can consult the book of Anderson [START_REF] Anderson | Hyperbolic geometry[END_REF] and we find the unpublished report of Parkkonen [START_REF] Parkkonen | Hyperbolic geometry[END_REF] very convenient. As above, the choice of the point 0 is irrelevant, let us choose for instance the center of the Euclidean ball on which is imposed the classical hyperbolic metric. Let λ be the Riemannian distribution on S n and σ be the corresponding hypersurface volume. Denote by Bp0, rq the closed ball in H n centered at 0 and of radius r ě 0. Up to a factor, we have λpBp0, rqq "

ż r 0 sinh n´1 puq du (18) σpBBp0, rqq " sinh n´1 prq (19)
From these formulas (and even only from [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF], since ( 18) is already a consequence of ( 19)), one can develop the same arguments as in the spherical situation, replacing the trigonometric functions by their hyperbolic counter-parts, to get the following results. A solution pBp0, R t qq tě0 of the radial evolution equation ( 3), is given by dR t " ? 2dB t `ˆ2 sinh n´1 pR t q JpR t q ´pn ´1q cothpR t q ˙dt where J : R `Q r Þ Ñ ş r 0 sinh n´1 puq du. In particular, for the hyperbolic plane (n " 2), we get

dR t " ? 2dB t `2 `coshpR t q sinhpR t q dt
Again, 0 is an entrance boundary for pR t q tě0 and we can make it start from 0, namely we can let pBp0, R t qq tě0 start from t0u. From this initial point, the process pλpR θt qq tě0 is a Bessel process of dimension 3 starting from 0, where the time change pθ t q tě0 is defined by

@ t ě 0, ż θt 0 1 sinh 2n´2 pJ ´1pR s qq ds " 2t
where J ´1 is the inverse mapping of J : R `Ñ R `. This is obtained through computations similar to those of Subsection 2.2 or as a consequence of Theorem 5.

Smooth initial conditions

After proving Theorem 3, we will show how to solve (3) for small times, when the initial domain has a smooth boundary. It will provide a solution of the martingale problem associated to L, thus showing Theorem 4.

As announced, we begin by the

Proof of Theorem 3

Consider R tpx, yq P R 2 : y ą 0u and the mapping

f : R Q px, yq Þ Ñ x y
For any f P C 8 pV q, we have Λrf s " fpF f , F 1 q, so that Λrf s P D.

It follows that

LrΛrf ss "

1 F 1 LrF f s ´Ff F 2 1 LrF 1 s ´2 F 2 1 Γ L rF f , F 1 s `2F f F 3 1 Γ L rF 1 , F 1 s
which can be rewritten under the form

F 1 LrΛrf ss " LrF f s ´2 F 1 Γ L rF f , F 1 s `Ff ˆ2 F 2 1 Γ L rF 1 , F 1 s ´1 F 1 LrF 1 s
Ẇe compute, for any D P D, with C BD, ν ν C and σ σ C ,

LrF 1 spDq " ż D Lr1s dµ `2 µpCq µpDq ż C 1 dµ " 2 µpCq 2 µpDq Furthermore, remark that Γ L rF 1 , F 1 spDq " ˆżC 1 dµ ˙2 " µpCq 2
so taking into account that F 1 pDq " µpDq, we get

2 F 2 1 Γ L rF 1 , F 1 s ´1 F 1 LrF 1 s " 0
Thus, we have

F 1 LrΛrf sspDq " LrF f spDq ´2 F 1 Γ L rF f , F 1 spDq " ż D Lrf s dµ `2 µpCq µpDq ż f dµ ´2µpCq µpDq ż C f dµ " ż D Lrf s dµ
and we conclude to the announced intertwining relation

LrΛrf ss " F Lrf s F 1
In the above proof, Definition (5) was helpful. Nevertheless to understand the dynamic of the domains generated by L, it is preferable to resort to [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF], so let us show its equivalence with [START_REF] Diaconis | Strong stationary times via a new form of duality[END_REF]. It amounts to check that for any D P D and any f P C 8 pV q, we have

ż D Lrf s dµ " ż C x∇f, ν C y `xβ, ν C y f dµ (20)
This equality is based on the integration by parts formula (Stokes' theorem), stating that for any smooth vector field v on V , we have

ż D divpvq dλ " ż C xv, νy dσ (21)
Indeed, we have [START_REF] Miclo | Isoperimetric stability of boundary barycenters in the plane[END_REF].

ż D Lrf s dµ " ż D p f `x∇U `β, ∇f yq exppU qdλ " ż D divpexppU q∇f q `xexppU
Now that we know that L satisfies the wanted intertwining relation with L, given D 0 P D, we would like to construct a Markov process pD t q tPr0,τq starting from D 0 and whose generator is L, where τ will be a positive stopping time, in a first step. To do so, we come back to the radial evolution equation (3) that we reinterpret under the heuristic D-valued stochastic differential equation

dD t " V 1 pD t q ˆ?2dB t `2 µpC t q µpD t q dt ˙`V 2 pD t q dt ( 22
)
where V 1 and V 2 are "vector fields" on D. This formulation will enable us to adapt the Doss-Sussman method [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF][START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF] to this infinite dimensional setting to construct a solution to the martingale problem associated to the generator L and to the initial position D 0 , at least for small times. Before explaining in general what we mean by a vector fields on D, we study the flow generated by V 1 , which is very simple to describe. For any r P R, denote ΨpD, rq

$ & % tx P V : dpx, Dq ď ru , if r ą 0 D
, if r " 0 tx P D : dpx, D c q ě ´ru , if r ă 0 [START_REF] Motoo | Proof of the law of iterated logarithm through diffusion equation[END_REF] where we recall that for any subset A Ă D and x P V , dpx, Aq inftdpx, yq : y P Au with d the Riemannian distance on V . It is easy to realize that the family pΨpD, rqq rPR does not behave well for some r P R: it does not stay in D and does not satisfy the flow property (see Remark 9 below). So we are going to restrict the parameter r to a convenient open segment containing 0.

For any x P V and v P T x V , let pexp x prvqq rPR stand for the geodesic flow whose position and speed at time 0 are x and v. By our assumption of completeness on V , these geodesic flows are defined for all times. For any r P R, define the mapping

ψ C,r : C Q x Þ Ñ exp x prν C pxqq (24)
Define R `pDq " inftr P p0, `8q : ψ C,r is not a diffeomorphism on its imageu (25) R ´pDq " ´inftr P p0, `8q : ψ C,´r is not a diffeomorphism on its imageu [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF] Due to the existence of a normal tubular neighborhood around the compact set C, we have that R `pDq ą 0 and R ´pDq ă 0. The interest of the segment pR ´pDq, R `pDqq is summarized as follows:

Proposition 8 Let D P D be given. For any r P pR ´pDq, R `pDqq, we have

BΨpD, rq " ψ C,r pCq " $ & % tx P D c : dpD, xq " ru , if r ą 0 C
, if r " 0 tx P D : dpD c , xq " ´ru , if r ă 0 [START_REF] Shiga | Bessel diffusions as a one-parameter family of diffusion processes[END_REF] showing that ΨpD, rq P D.

Furthermore, for any r, r 1 P pR ´pDq, R `pDqq such that r `r1 P pR ´pDq, R `pDqq, the "semi-group property" holds: ΨpD, r `r1 q " ΨpΨpD, rq, r 1 q " ΨpΨpD, r 1 q, rq Proof The above result is certainly standard, even we were not able to find a corresponding reference.

For the first assertion, we begin by considering the case r P p0, R `pDqq. For any x P ΨpD, rqzD, there exists y P C such that dpx, yq " dpx, Dq P p0, rs. Let us check that x " ψ C,dpx,yq pyq. Denote pγpsqq sPr0,dpx,yqs a unitary minimizing geodesic going from y to x. There exists v P T y V with }v} " 1 such that γpsq " exp y psvq for all s P r0, dpx, yqs. If v is not orthogonal to T y C, then for small s ą 0, we could find y s P C with dpy s , γpsqq ă dpy, γpsqq, contradicting the minimizing property of y, since we would get dpx, yq " dpy, γpsqq `dpγpsq, xq ą dpy s , γpsqq `dpγpsq, xq ě dpx, y s q. If v was directed toward the interior of D, we would also end up with a contradiction, by considering the last time s P p0, dpx, yqq such that γpsq P D. It follows that v " ν C pyq, showing that x " ψ C,dpx,yq pyq. We furthermore get such a point y P C is unique, otherwise we would be in contradiction with the fact that ψ C,dpx,Dq is injective. Conversely, if s P p0, rs and y P C, then x ψ C,s pyq P ΨpD, sq, with dpx, Dq ď dpx, yq ď s. Thus we have the description

@ r P p0, R `pDqq, ΨpD, rq " D ď sPp0,rs ψ C,s pCq
Let us show that all the sets of the r.h.s. are disjoint. First we prove by contradiction that @ s P p0, rs, D X ψ C,s pCq " H

So assume that ψ C,s pxq P D, for some x P C. Replacing s by inftt ą 0 : ψ C,t pxq P Du, which is still positive, because ψ C,t pxq does not belong to D for t ą 0 small enough, we can assume that ψ C,s pxq P C. Consider the mapping φ : r0, ss Q t Þ Ñ dpψ C,t pxq, Cq. We have seen above that for t ą 0 small enough, we have φptq " t. Since φpsq " 0, let u inftt ą 0 : φptq " tu, which belongs to p0, sq. Note that for t P r0, uq, the directing normal vector d dt ψ C,t pxq is orthogonal to the tangent space of ψ C,t pCq at ψ C,t pxq, otherwise for v P pt, uq, we could find a shortest way from ψ C,v pxq to ψ C,t pCq than the one given by the geodesic pψ C,w pxqq wPrt,vs and it would follow that dpψ C,v pxq, Cq ă v. The tangent space of ψ C,t pCq at ψ C,t pxq coincides with the image of T x C by T ψ C,t pxq, by the fact that ψ C,t is a diffeomorphism on its image. Letting t go to u, we get the directing normal vector d dt ψ C,t pxq ˇˇt"u is still orthogonal to the tangent space of ψ C,u pCq at ψ C,u pxq. As above, this property insures us that for ą 0 small enough,

dpψ C,u` pxq, ψ C,u pCqq " (29)
namely either dpψ C,u` pxq, Cq " u ` or dpψ C,u` pxq, Cq " u ´ . The first alternative is forbidden by the definition of u. For the second alternative, we get, for ą 0 small enough, ψ C,u` pxq " ψ C,u´ pxq belongs to ψ C,u´ pCq, thus we can find y P Cztxu with ψ C,u` pxq " ψ C,u´ pyq. If follows from (29) that ψ C,u pxq " ψ C,u pyq, in contradiction with the injectivity of ψ C,u . This ends the proof of [START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF]. The proof that for s " s 1 P p0, rs, we have ψ C,s pCq X ψ C,s 1 pCq " H is similar. Indeed, if this equality was not true, then one would be able to find again x P C and t P p0, rs such that dpψ C,t pxq, Cq ą t. We end up with the "foliation"

@ r P p0, R `pDq, ΨpD, rq " D ğ sPp0,rs ψ C,s pCq (30)
From this decomposition and the continuity of C ˆp0, R `pDqq Q px, sq Þ Ñ ψ C,s pxq, we deduce that for r P p0, R `pDqq, BΨpD, rq " ψ C,r pCq " tx P D c : dpD, xq " ru

The analogous relations when r P pR ´pDq, 0q are obtained in a similar way, taking into account that @ r P pR ´pDq, 0q, ΨpD, rq " Dz ¨ğ sPrr,0q ψ C,s pCq ' (31)

The semigroup property is also a consequence of (30) and (31), taking into account that for r, r 1 as in the above proposition, we have

ψ C,r`r 1 " ψ C,r ˝ψC,r 1 " ψ C,r 1 ˝ψC,r
(remarking that for any x P C and r P pR ´pDq, R `pDqq, we have T x ψ C,r rν C pxqs " ν ΨpC,rq pψ C,r pxqq).

Remark 9

The semi-group property of Corollary 8 is no longer necessarily true if the conditions on r, r 1 P R are not satisfied. Consider first the following (non connected) example: let D be the union of the open balls Bpp0, 0q, 3q and Bpp0, 5q, 1q. Then we have ΨpD, ´2q " Bpp0, 0q, 1q and ΨpBpp0, 0q, 1q, 2q " Bpp0, 0q, 3q " D. This example can be modified into a connected one by joining Bpp0, 0q, 3q and Bpp0, 5q, 1q through the open rectangle r0, 5s ˆr´1, 1s. The boundary of the resulting domain D is not smooth, nevertheless, the definition (23) makes sense. The boundary BΨpD, rq makes an "irreversible transition" at r " ´1.

From now on, for r P pR ´pDq, R `pDqq, denote by ΨpC, rq the set described in [START_REF] Shiga | Bessel diffusions as a one-parameter family of diffusion processes[END_REF]. For given D P D, the family pΨpC, rqq rPpR ´pDq,R `pDqq is the solution of the normal flow equation, which can be written under the radial form " ΨpC, 0q " C @ r P pR ´pDq, R `pDqq, @ x P ΨpC, rq, B r x " ν ΨpC,rq pxq

where the points of the boundaries are pushed according to the outward normal.

For our purposes, it is convenient to look at this set-valued evolution through our elementary observables:

Lemma 10 Let D P D and f P C 8 pV q be fixed. The mapping pR ´pDq, R `pDqq Q r Þ Ñ F f pΨpD, rqq P R is C 2 and for any r P pR ´pDq, R `pDqq, we have

B r F f pΨpD, rqq " ż ΨpC,rq f dµ B 2 r F f pΨpD, rqq " ż ΨpC,rq @ ∇f, ν ΨpC,rq D dµ `żΨpC,rq p @ ∇U, ν ΨpC,rq D `ρΨpC,rq qf dµ
To simplify the notation, when the set C will be clear from the context (e.g. coming from the domain of integration), we will write σ, ν and ρ instead of σ C , ν C and ρ C , convention which was already adopted for µ. We will also need to differentiate Ψ with respect to the first variable D P D. We must first give a meaning to the underlying notion of differentiation in D.

Consider a family pG s q sPr0,S `q taking values in D, for a real number S `ą 0. We say this family is strongly continuous in a neighborhood of s P r0, S `q if there exist a neighborhood N s of s in r0, S `q and a continuous mapping ϕ s : N s ˆBG s Ñ V such that for any u P N s , the function BG s Q x Þ Ñ ϕ s pu, xq is a homeomorphism between BG s and BG u and if ϕ s ps, ¨q is the identity mapping. In this statement, the boundaries BG s , for s P r0, S `q are endowed with the topology inherited from that of V . Similarly, these boundaries will be endowed below with the smooth differentiable structure inherited from V as smooth submanifolds. The family pG s q sPr0,S `q is said to be strongly continuous on r0, S `q, if for any s P r0, S `q, it is strongly continuous in a neighborhood of s.

Remark 11 Let d be the Hausdorff metric on the compact subsets of V . It endows D with a metric structure. The strong continuity defined above implies the continuity for the Hausdorff metric, but the converse is not always true, as it is illustrated by the following picture:

Note

that the restrictions to D of the mappings defined in [START_REF] Gallot | Riemannian geometry. Universitext[END_REF] and ( 11) are strongly continuous. By analogy, we say the family pG s q sPr0,S `q is strongly smooth in a neighborhood of s P r0, S `q if there exist a neighborhood N s of s in r0, S `q and a smooth mapping ϕ s : N s ˆBG s Ñ V such that for any u P N s , the function BG s Q x Þ Ñ ϕ s pu, xq is a diffeomorphism between BG s and BG u and ϕ s ps, ¨q is the identity mapping. The family pG s q sPr0,S `q is then said to be strongly smooth if it is strongly smooth in the neighborhood of any s P r0, S `q. For such a family, consider for any s P r0, S `q and x P BG s , the vector

X BGs pxq B u ϕ s pu, xq| u"s
The T V -valued vector field X BGs on BG s enables to describe the infinitesimal evolution of G s via a formula similar to (32) @ s P r0, S `q, @ x P BG s , B s x " X BGs pxq This description is not unique, because the mappings ϕ s pu, ¨q are not unique: they can be composed by diffeomorphisms of BG u , depending on s and (smoothly) on u. Indeed, as already mentioned, the discussion of Section 1.3 of Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF] shows that for x P BG s , only the radial part α BGs pxq xX BGs pxq, ν BGs pxqy is unique. Furthermore, it is possible to choose the mappings ϕ s in such a way so that @ s P r0, S `q, @ x P BG s , X BGs pxq " α BGs pxqν BGs pxq and the function α is continuous in the sense that if the sequences ps n q nPN in r0, S `q and px n q nPN , taking values respectively in pBG sn q nPN , are converging toward s P r0, S `q and x P BG s , then lim nÑ8 α BGs n px n q " α BGs pxq. The family pG s q sPr0,S `q can thus be described more intrinsically as a solution of the radial equation equation @ s P r0, S `q, @ x P BG s , B s x " α BGs pxqν BGs pxq

This formula enables us to identify the "tangent space" T D D at D P D with the space C 8 pCq of real smooth functions on C (of the form α C with the above notation). At least it appears that

T D D Ă C 8 pCq.
Conversely, given α P C 8 pCq, we will see in Remark 16 how to construct a strongly smooth family pG s q sPr0,S `q such that " G 0 " D @ x P BG 0 , B s x| s"0 " αpxqν BG 0 pxq (34)

This shows that C 8 pCq Ă T D D.

Following the traditional definition in differential geometry, we say that a mapping Φ : D Ñ D is strongly smooth if any strongly smooth family pG s q sPr0,S `q is transformed by Φ into a strongly smooth family, i.e. pΦpG s qq sPr0,S `q is smooth (to simplify the terminology, from now on, smooth means strongly smooth). Then there exists a vector field r α on pΦpG s qq sPr0,S `q such that @ s P r0, S `q, @ x P BΦpG s q, B s x " r α BΦpGsq pxqν BΦpGsq pxq Fix s P r0, S `q. It is not difficult to see that the function r α BΦpGsq depends on α satisfying (33) only through α BGs . For fixed D P D, consider any smooth and D-valued family pG s q sPr0,S `q with 0 P r0, S `q and G 0 " D. Let α be associated with pG s q sPr0,S `q as in (33). The linear functional transforming α C into r α BΦpDq , as above, is called the tangent mapping T D Φ of Φ at D. α on D in order to be able to solve locally in time (34) to get smooth families pG s q sPr0,S `q and p r G s q sPr0, r S `q. Replace S `by S `^r S `. Assuming that α did not vanish identically on BD, we can furthermore impose that S `is small enough so that r0, S `q Q s Þ Ñ G s is one-to-one. It enables us to define Φ on tG s : s P r0, S `qu via ΦpG s q " r G s , for all s P r0, S `q. Then we get T D Φrαs " r α. To go further would require a better understanding of the neighborhood of D in D. With all these preliminaries at our disposal, we can now compute the tangent mapping T D Ψp¨, rq for r P pR ´pDq, R `pDqq. Rigorously, for given r P R, the mapping Ψp¨, rq is not defined on the whole set D but only on the subset

D r tD P D : r P pR ´pDq, R `pDqqu ( 35
)
This subset is open for the strong topology alluded before (but not in the Hausdorff topology, see Remark 11), so that the notion of tangent mapping can be extended to this setting (as soon as D r " H).

The tangent mapping T D Ψp¨, rq is among the simplest possible ones:

Lemma 13 Let D P D and r P pR ´pDq, R `pDqq be given. For any α P C 8 pCq and x P C, we have

T D Ψp¨, rqrαspxq " αpψ ´1 C,r pxqq
where ψ ´1 C,r : ΨpC, rq Ñ C is the inverse mapping of the function ψ C,r defined in [START_REF] Parkkonen | Hyperbolic geometry[END_REF].

Proof

Let α P C 8 pCq be given, extend it smoothly on V and solve (34) for ą 0 small enough. For x P C and s P p´ , q, denote ϕpx, sq x s and A s tϕpx, sq : x P Cu. According to the previous discussion, to get the wanted result, we just need to check that for any x P C, the part of B s ψ As,r pϕpx, sqq| s"0 which is (outwardly) normal to ΨpA s , rq is equal to αpxq, namely that @ x P C, @ B s ψ As,r pϕpx, sqq| s"0 , ν ΨpC,rq pψ C,r pxqq D ψ C,r pxq " αpxq (36) Denote @ t P r0, rs, J t B s ψ As,t pϕpx, sqq| s"0 so that pJ t q tPr0,rs is a vector field over the geodesic pγptqq tPr0,rs pψ C,t pxqq tPr0,rs . For all s P p´ , q, pψ C,t px s qq tPr0,rs is a geodesic, it follows that pJ t q tPr0,rs is a Jacobi fields (cf. for instance Proposition 3.45 from the book of Gallot, Hulin and Lafontaine [START_REF] Gallot | Riemannian geometry. Universitext[END_REF], whose Chapter 3 serves as a reference for all the following considerations). Thus pJ t q tPr0,rs is defined by its initial conditions Jp0q and J 1 p0q, where the prime corresponds to the covariant derivative with respect to t, and by the evolution J 2 " ´RpJ, 9 γq 9 γ, where R is the Riemannian curvature tensor. To prove (36) amounts to show that the mapping r0,

rs Q t Þ Ñ xJptq, 9
γptqy γptq is constant. The covariant derivative is constructed so that the scalar product is left invariant, so that since the p0, 4q-curvature tensor R is anti-symmetric in its last two vector variables (as well as in first two vector variables). Thus, to get the wanted result, we just need to check that J 1 p0q is orthogonal to 9 γp0q " ν C pxq. From the first equality of Proposition 3.29 of Gallot, Hulin and Lafontaine [START_REF] Gallot | Riemannian geometry. Universitext[END_REF] (applied with the commutating vector fields X " B s and J " B t on p´ , q ˆr0, rs parametrized by ps, tq), it appears that J 1 p0q coincides with the covariant derivative with respect to s of the tangent vectors of the geodesic pψ C,t px s qq tPr0,rs , at s " 0 and t " 0. The latter tangent vectors are unitary, so their covariant derivatives are orthogonal to them. Thus at s " 0 and t " 0 we get xJ 1 p0q, 9 γp0qy x " 0, ending the proof of (36).

@ t P r0,
We deduce the differentiation of our favorite observables.

Corollary 14

In the setting of Lemma 13, let be given f P C 8 pV q and pG s q sPr0,S `q with G 0 " D and α BG 0 " α (in the sense of (33)). We have

d ds F f pΨpG s , rqq ˇˇˇs "0 " ż ΨpC,rq f pxqαpψ ´1 C,r pxqq µpdxq

Proof

As in the first part of the proof of Lemma 10, we get

d ds F f pG s q ˇˇˇs "0 " ż C f pxqαpxq µpdxq
Taking into account Lemma 13, the announced result follows from this formula, with pG s q sPr0,S `q replaced by pΨpG s , rqq sPr0,S `q.

A famous example of radial evolution of the type (33) is the mean curvature flow: @ s P r0, S `q, @ x P BG s , B s x " ´ρBGs pxqν BGs pxq where G 0 P D is given and r0, S `q is the maximum interval on which this flow remains in D (there are various ways to define the mean curvature flow beyond the times when it gets out of D, see e.g. Chapter 1 of the book of Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF]). When V " R 2 endowed with its usual Riemannian structure, it is possible to compute explicitly the image of the mean curvature vector field ρ by the tangent applications to the normal flow Ψ, see Subsection 5.1. In general, it is more difficult (see nevertheless Remark 49 for the usual Riemannian structure on V " R n ), since the curvature of V will enter into the game. The arguments of Section 1.5 of Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF] can be adapted to get existence and uniqueness of the solutions pG s q sPr0,S `q to the radial evolution equations of the form @ s P r0, S `q, @ x P BG s , B s x " p´ρ BGs pxq `xbpxq, ν BGs pxqy x `apxqqν BGs pxq

where r0, S `q is a small enough interval containing 0, where G 0 is a given element from D and where a and b are respectively a smooth function and a smooth vector field on V . The obtained solution pG s q sPr0,S `q is a smooth family. The underlying idea is to consider again the parametrization pr ´, r `q ˆG0 Q pr, xq Þ Ñ ψ G 0 ,r pxq of a tubular neighborhood of G 0 , where pr ´, r `q is a small neighborhood of 0. Then one looks for a mapping r0, S `q ˆG0 Q ps, xq Þ Ñ yps, xq, whose image is included into the tubular neighborhood ψ G 0 ,pr ´,r `qpG 0 q and which is such that for any s P r0, S `q and any x P G 0 where ϕ 0 : G 0 Ñ R n is the inclusion map. Then writing yps, xq " ψ G 0 ,f ps,xq pxq, for all ps, xq P r0, S `q Ĝ0 , we end up with the quasi-linear parabolic equation with respect to f : for any

x P G 0 , $ & % f p0, xq " 0 @ s P r0, S `q, B s f ps, xq " G 0 ,s f ps, xq `Hpx, f ps, xq, pB x i f ps, xqq iP n´1 , pB x i pB x j f ps, xqq 2 q i,jP n´1 q (38)
where H is a smooth mapping on R `ˆR n´1 ˆRpn´1q 2 and where G 0 ,s is the Laplacian relatively to the Riemannian structure on G 0 obtained by pulling back through the diffeomorphism G 0 Q x Þ Ñ ψ G 0 ,f ps,xq pxq the Riemannian structure on G s inherited from that of V . Note that H will be independent from the chart in which we compute B x i f ps, xq and B x i B x j f ps, xq.

Before going further, let us explain how to get (38) from (37), when V " R n . In this case we have yps, xq " ψ G 0 ,f ps,xq pxq " ϕ 0 pxq `f ps, xqν 0 pxq and note that for f small and smooth, yps, .q will be a diffeomorphism, with f p0, xq " 0 and B x i f p0, xq " 0 and B x i B x j f p0, xq " 0. We compute the equation satisfied by f ps, xq such that yps, xq is a solution of (37). Taking into account Corollary 1.3.5 in Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF], up to a reparametrization, the evolution of G s " yps, G 0 q is characterized by it's normal evolution, namely xB s yps, xq, ν BGs pyps, xqqy yps,xq . Let us compute the pullback metric at x P G 0 , gps, xq " yps, .q ˚g|BGs , where g is the canonical metric in V . In a local chart of G 0 , px i q iP n´1 at x P G 0 , we have:

g i,j ps, xq " xB x i yps, xq, B x j yps, xqy " xB x i ϕ 0 `Bx i f ps, xqν 0 pxq `f ps, xqB x i ν 0 pxq, B x j ϕ 0 `Bx j f ps, xqν 0 pxq `f ps, xqB x j ν 0 pxqy " xB x i ϕ 0 `f ps, xqB x i ν 0 pxq, B x j ϕ 0 `f ps, xqB x j ν 0 pxqy `Bx i f ps, xqB x j f ps, xq
Using Gauss-Weingarten equation, namely:

B x i ν 0 pxq " h i,l p0, xqg l,k p0, xqB x k ϕ 0 pxq
where h i,l p0, xq is the second fundamental form of G 0 at x, and pg l,k p0, xqq l,k is the inverse of the metric pg l,k p0, xqq l,k , and we use the convention that every repeated lower indices and upper indices are considered as a sum, as in the whole paper. We get,

g i,j ps, xq " g i,j p0, xq `2f ps, xqh i,j p0, xq `f 2 ps, xqh i,l g l,m h j,m p0, xq `Bx i f ps, xqB x j f ps, xq
Using again Gauss-Weingarten equation, and since xνps, xq, B x i yps, xqy " 0 we have

h i,j ps, xq " ´xνps, xq, B x i B x j yps, xqy " ´xνps, xq, B x i B x j ϕ 0 `Bx i B x j f ν 0 pxq `Bx j f B x i ν 0 pxq `Bx i f B x j ν 0 `f B x i B x j ν 0 pxqy " ´Bx i B x j f xνps, xq, ν 0 pxqy `p H i,j px, f ps, xq, pB x l f ps, xqq lP n´1 q
where νps, xq is the exterior normal vector of G s at the point yps, xq, and h i,j ps, xq is the second fundamental form of G s at yps, xq in the basis pB x i yps, xqq iP n´1 of it's tangent space and p H is a smooth function when the two last argument are small enough. We also have B s yps, xq " B s f ps, xqν 0 pxq and ρ BGs pyps, xqq " g i,j ps, xqh i,j ps, xq, note that this quantity is independent of the chart. If we write :

ρ BGs pyps, xqq " g i,j ps, xqh i,j ps, xq " ´gi,j ps, xqB x i B x j f xνps, xq, ν 0 pxqy `gi,j ps, xq p H i,j px, f ps, xq, pB x l f ps, xqq lP n´1 q " ´gi,j ps, xq `Bx i B x j f ´Γk i,j ps, xqB x k f ps, xq ˘xνps, xq, ν 0 pxqy `gi,j ps, xq `p H i,j px, f ps, xq, pB x l f ps, xq lP n´1 q ´Γk i,j ps, xqB x k f ps, xqxνps, xq, ν 0 pxqyq " ´ G 0 ,s f xνps, xq, ν 0 pxqy `q Hpx, f ps, xq, pB x l f ps, xqq lP n´1 , pB x l pB x k f ps, xqq 2 q l,kP n´1 q
where Γ k i,j ps, xq is the Christoffel for the metric gpsq, this quantity depends on the derivative of g and thus on the second derivative of f , but only via pB x l pB x k f ps, xqq 2 q l,kP n´1 . Furthermore, as we can see below, νps, xq depends on the derivative of f up to order one. Note that since ρ BGs ps, xq and G 0 ,s f are independent on the choice of the chart, the same is true for q Hpx, f ps, xq, pB x l f ps, xqq lP n´1 , pB x l pB x k f ps, xqq 2 q l,kP n´1 qq. So if yps, xq is a solution of (37) then after taking bracket with the normal vector we get:

B s f ps,
" G 0 ,s f xνps, xq, ν 0 pxqy `r Hpx, a, b, f ps, xq, pB x l f ps, xqq lP n´1 , pB x l pB x k f ps, xqq 2 q l,kP n´1 q (39)
where r H is independent on the choice of the chart. Furthermore, since pv i ps, xqq iP n´1 p ? g i,l ps, xqB x l yps, xqq iP n´1 is an orthonormal basis of the tangent space of G s at yps, xq, the vector ν 0 pxq ´řiP n´1 xν 0 pxq, v i ps, xqyv i ps, xq is orthogonal to this tangent space. Let us compute it, taking into account that xν 0 pxq, B x l ϕ 0 pxqy " 0 and xν 0 pxq, B x l ν 0 pxqy " 0:

ν 0 pxq ´ÿ iP n´1 xν 0 pxq, v i ps, xqyv i ps, xq " ν 0 pxq ´ÿ iP n´1 C ν 0 pxq, ÿ lP n´1 ? g i,l ps, xqB x l yps, xq G ÿ kP n´1 ? g i,k ps, xqB x k yps, xq " ν 0 pxq ´ÿ iP n´1 ÿ lP n´1 ? g i,l ps, xqxν 0 pxq, B x l yps, xqy ÿ kP n´1 ? g i,k ps, xqB x k yps, xq " ν 0 pxq ´ÿ iP n´1 ÿ lP n´1 ? g i,l ps, xqB x l f ps, xq ÿ kP n´1 ? g i,k ps, xqB x k yps, xq " ν 0 pxq ´ÿ lP n´1 ÿ kP n´1 ÿ iP n´1 ? g i,l ps, xq ? g i,k ps, xqB x l f ps, xqB x k yps, xq " ν 0 pxq ´ÿ lP n´1 ÿ kP n´1 g kl ps, xqB x l f ps, xqB x k yps, xq
In particular, this vector is different to zero for f and ∇f small enough and we get then

νps, xq " ν 0 pxq ´Bx i f ps, xqg i,j ps, xqB x j pϕ 0 pxq `f ps, xqν 0 pxqq }ν 0 pxq ´Bx i f ps, xqg i,j ps, xqB x j pϕ 0 pxq `f ps, xqν 0 pxqq}
It follows equally that xνps, xq, ν 0 pxqy is different to zero for f and ∇f small enough, and thus dividing (39) by xνps, xq, ν 0 pxqy, we get (38) for a smooth function H deduced from the previous computations.

When s, f ps, xq and ∇ G 0 ,s f ps, xq are small, the implicit function theorem enables us to write (38) under the form considered in Appendix A of Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF], due to the strict ellipticity of the operator G 0 ,s on G 0 and to the fact that

pB x i pB x j f ps, xqq 2 q i,jP n´1 " 2pB x j f ps, xqpB x i B x j f ps, xqqq i,jP n´1
As shown by Appendix A of Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF], such quasi-linear parabolic equations admit a unique solution on a small time interval containing 0, so this existence and uniqueness result holds for (38). It would also be possible to put in front of the term ρ BGs pxq of (37) a positive quantity depending smoothly on x.

Remark 15

We have written in a natural way the leading term of ρ BGs in terms of the Laplacian for the metric gpsq. Unfortunately the equation we will need will not be exactly of this form, because we will have an additional stochastic term, carefully studied in Section 4. For the short time existence, we will prefer to write this leading term in terms of a fixed manifold with a fixed metric as in (62) in Subsection 4.1 .

Remark 16 Let us come back to the search of a smooth family pG s q sPr0,S `q satisfying (34), where α P C 8 pG 0 q is given. First extend ρ BG 0 `α from BG 0 to V , to obtain a smooth function a P C 8 pV q coinciding with ρ BG 0 `α on BG 0 . Next define for any D P D,

@ x P C, α C pxq " ´ρC pxq `apxq
The radial evolution equation @ s P r0, S `q, @ x P BG s , B s x " α BGs pxqν BGs pxq (40) is of the form (37) and so admits a unique solution for small enough intervals r0, S `q. Restricting the above equation to s " 0 shows that pG s q sPr0,S `q solves (34). This construction seems particularly cumbersome, it would be more natural to extend α from BG 0 to V to get a smooth function c P C 8 pV q and to solve the radial evolution equation @ s P r0, S `q, @ x P BG s , B s x " cpxqν BGs pxq (41)

Unfortunately, doing so, we end up with a Hamilton-Jacobi equation (see e.g. Chapter 3 of Evans [START_REF] Evans | Partial differential equations[END_REF]) instead of the quasi-linear parabolic equation (38). One would then be led to investigate if the usual conditions for existence and uniqueness of the solutions to the Hamilton-Jacobi equations are satisfied and thus to describe more precisely the function H appearing in (38), but this is not so nice.

The normal flow equation (32), corresponding to c " 1, was simple to solve (in both direction of the time, contrary to the above quasi-linear parabolic equations), because the normal vectors are transported in a parallel way by the geodesic flows directed by these normal vectors.

Equations of the type (37) are adapted to our purposes: only considering the last vector field in [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF], i.e. the heuristic D-valued "ordinary" differential equation dD t " V 2 pD t qdt, amounts to solve the following modification of the mean curvature flow:

@ s P r0, S `q, @ x P BG s , B s x " ´ρb BGs pxqν BGs pxq (42)
where

@ D P D, @ x P C, ρ b C pxq ρ C pxq `x∇U pxq ´βpxq, ν C pxqy x (43)
(despite the b in supscript, remember that b " ∇U `β and not ∇U ´β, as the above formula could suggest). Let D 0 P D be given, as well as pB t q tě0 a standard (one-dimensional) Brownian motion starting from 0. To solve [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF], we are looking for a stochastic D-valued evolution pD t q tPr0,τq , where τ ą 0 is a stopping time (wrt. to the filtration generated by the Brownian motion), such that

@ t P r0, τq, @ x P C t , dx " ˆ?2dB t `2 µpC t q µpD t q dt ´ρb Ct pxqdt ˙νCt pxq (44)
where C t BD t .

To explain the Doss [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF] and Sussman [START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF] approach to such stochastic differential equations, it is helpful to first replace ? 2 dB t `2µpC t q{µpD t q dt by dξ t " ξ 1 t dt, where ξ : R `Ñ R is a given C 1 function with ξ 0 " 0. Still starting from D 0 , we would like to solve the radial evolution equation

@ t P r0, q, @ x P C t , B t x " `ξ1 t ´ρb Ct pxq ˘νCt pxq (45)
for some ą 0, without using the derivative pξ 1 t q tPr0, s . To do so, we begin by solving another radial evolution equation

" G 0 " D 0 @ t P r0, r q, @ x P BG t , B t x " α BGt,ξt pxqν BGt pxq (46)
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 ´ρb ΨpC,rq pψ C,r pxqq (47)

where ΨpC, rq was defined after Remark 9, taking into account ( 24), ( 25), ( 26), (35). Next, consider inftt P r0, r q : G t R D ξt u ą 0 (with the usual convention that " r if the set in r.h.s. is empty) and define

@ t P r0, q, D t ΨpG t , ξ t q
Let us check that this is indeed a solution of (45). First, we have ΨpG 0 , ξ 0 q " ΨpD, 0q " D 0 . Concerning the evolution, differentiate with respect to the first and second variables of Ψ to find @ t P r0, q, @ x P C t , B t x " pT Gt Ψp¨, ξ t qrα BGt,ξt spxq `ξ1 t q ν Ct pxq " `´ρ b Ct pxq `ξ1 t ˘νCt pxq as wanted, where we used Lemma 13. Denote h the mapping defined on D by

@ D P D, hpDq " 2 µpCq µpDq
For given D 0 P D and a C 1 function ζ : R `Ñ R, we are now looking for a solution, starting from D 0 , to the radial evolution equation @ t P r0, q, @ x P C t , B t x " `ζ1 t `hpD t q ´ρb Ct pxq ˘νCt pxq (48)

for some ą 0. Following computations similar to those presented above, we get a solution by taking, for t ą 0 small enough,

D t ΨpG t , ζ t `θt q (49)
where the R `ˆD-valued family pθ t , G t q tPr0, q , for ą 0 small enough, is a solution of the system starting from pθ 0 , G 0 q " p0, D 0 q and satisfying @ t P r0, q,

" d dt θ t " hpΨpG t , ζ t `θt qq @ x P BG t , B t x " α BGt,ζt`θt pxqν BGt pxq (50)
The formulations (49) and (50) do not require that the function ζ is differentiable. These remarks suggest to solve (44) by replacing pζ t q tě0 by p ? 2B t q tě0 in (49) and (50), up to the random time τ these constructions are allowed: τ will be a stopping time with respect to the filtration generated by the Brownian motion pB t q tě0 . This is the Doss [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF] and Sussman [START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF] method, adapted to our evolving domain framework.

So given D 0 P D, we are led to consider the following stochastic radial evolution equation system with respect to pθ t , G t q tPr0, q , starting with pθ 0 , G 0 q " p0, D 0 q: @ t P r0, q,

" d dt θ t " hpΨpG t , ? 2B t `θt qq @ x P BG t , B t x " α BGt, ? 2Bt`θt pxqν BGt pxq (51)
In Section 4, we show the existence of a solution of (46), where pξ t q tě0 " p ? 2B t q tě0 and the existence of a solution of (51). There, we will only consider the case V " R n`1 , the situation of a general manifold V is similar up to some modifications, which are straightforward from a conceptual point of view, but induce complicated notations.

Once (51) is solved, define as in (49),

@ t P r0, τq, D t ΨpG t , ? 2B t `θt q. ( 52
)
up to the stopping time τ until which this construction is permitted.

Let us now check that (52) provides a solution to the martingale problem presented in the introduction:

Theorem 17 The stopped stochastic process pD t q tPr0,τq , defined on the natural filtered probability space of the standard Brownian motion pB t q tě0 , is a solution to the stopped martingale problem associated to the generator pD, Lq and to the starting domain D.

Proof

Fix some f P C 8 pV q. On the set I tps, rq P R `ˆR : G s P D r u, consider the mapping

ps, rq Þ Ñ F f pΨpG s , rqq (53)
According to Lemma 10, this mapping is C 2 in the second variable. Concerning the first variable, note that for ps, rq P I, we have

@ x P BG s , B s x " ´ρb ΨpBGs,rq pψ BGs,r pxqqν BGs pxq (54)
From Lemma 13, we deduce that where we used [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF] and where pM t q tPr0,τq is a local martingale whose bracket is given by

@ x P ΨpBG s ,
@ t P r0, τq, xM y t " 2 ż t 0 Γ L rF f , F g spD s q ds
This description and the continuity of the trajectories r0, τq Q t Þ Ñ F f pD t q imply that pD t q tPr0,τq is a solution to the martingale problem associated to the generator pD, Lq (see e.g. the book of Bakry, Gentil and Ledoux [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]). Since D 0 " D, we conclude to the wanted result.

Remark 18 There are potentially other ways to use the Doss-Sussman approach. For instance, Equation ( 22) can be rewritten under the form

dD t " ? 2 V 1 pD t q dB t `r V 2 pD t q dt ( 57
)
where r V 2 pDq 2hpDqV 1 pDq `V2 pDq for any D P D. Similarly to (43) and (47), define

@ D P D, @ x P C, r ρ b C pxq ρ C pxq `x∇U pxq ´βpxq, ν C pxqy x ´hpDq @ r ą 0, @ D P D r , @ x P C, r α C,r pxq ´r ρ b ΨpC,rq pψ C,r pxqq
Next try to construct a family p r G t q tPr0, q (where ą 0 is a stopping time) such that

@ t P r0, q, @ x P B r G t , B t x " r α BGt, ? 2Bt pxqν BGt pxq
Contrary to (51), no auxilliary pθ t q tPr0, q is needed here, but the above equation is not really of the type (37), due to the isoperimetric ratio. Nevertheless, it should be possible to adapt to this situation the fixed point approach presented in Section 4. Once p r G t q tPr0, q has been constructed, consider

@ t P r0, τq, D t Ψp r G t , ? 2B t q with τ inftt P r0, q : r G t R D ? 2Bt u
Then the stopped stochastic process pD t q tPr0,τq , defined on the natural filtered probability space of the standard Brownian motion pB t q tě0 , is a solution to the martingale problem associated to the generator pD, Lq and to the starting domain G 0 . We preferred to present how to solve [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF], because the flows associated to V 1 and V 2 are quite famous (at least when ∇U " β " 0) and well-investigated. But maybe the flow associated to the radial equation

@ x P C t , B t x " phpD t q ´ρCt pxqq ν Ct pxq
is also a natural object to study. In Subsection 5.2, we will check in the homogeneous setting of Section 2 that this alternative Doss-Sussman approach should be preferred to the one considered in the proof of Theorem 17. 4

Existence of a stochastic modified mean curvature flow

This section presents the quite technical proofs of the existence of regular solutions to (46) and (51), respectively the following subsections. As announced before Theorem 17, we only deal with V " R n`1 to avoid complicated notations.

We begin by recollecting our notations: D is the set of non-empty, compact and connected domains D in V , which coincide with the closure of their interior and whose boundary C BD is smooth. The exterior normal vector ν C and the mean curvature ρ C are defined on C. Recall we were given a function U P C 8 pV q and a smooth vector field β satisfying divpexppU qβq " 0, to which is associated the smooth vector field b ∇U `β. Denote µ exppU qλ, the measure admitting the density exppU q with respect to the Riemannian measure λ (when µ gives a finite weight to V , it is normalized into a probability measure, which amounts to add a constant to U ). The interest of µ is to be reversible for the operator L `b. We associate to the boundary C the pdimpV q ´1q-Hausdorff measure µ C coming from µ, namely admitting the density exppU q with respect to the usual Riemannian pdimpV q ´1q-Hausdorff measure. We also distort ρ C by introducing the modified mean curvature ρ b C defined by

@ x P C, ρ b C pxq ρ C pxq `x∇U pxq ´βpxq, ν C pxqy x
Let D 0 P D be given, as well as pB t q tě0 a standard real Brownian motion starting from 0. We are looking for a stochastic D-valued evolution pD t q tPr0,τq , where τ ą 0 is a stopping time, such that @ t P r0, τq, @ x P C t , dx " ´?2dB t `2hpD t qdt ´ρb Ct pxqdt ¯νCt pxq (44

)
where

@ D P D, hpDq 2 µpCq µpDq
Resorting to the Doss [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF] and Sussman [START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF] method, we are led to solve consecutively:

• The deterministic radial equation in pG t q tPr0,r q : " G 0 " D 0 @ t P r0, r q, @ x P BG t , B t x " α BGt,ξt pxqν BGt pxq

where R `Q t Þ Ñ ξ t P R is assumed to be α-Hölder regular with α P p0, 1{2q, r is small enough and @ r ą 0, @ D P D r , @ x P C, α C,r pxq ´ρb ΨpC,rq pψ C,r pxqq with for any r P R,

ψ C,r : C Q x Þ Ñ exp x prν C pxqq P V ΨpC, rq
tψ C,r pxq : x P Cu D r tD P D : r P pR ´pDq, R `pDqqu R `pDq inftr P p0, `8q : ψ C,r is not a diffeomorphism on its imageu R ´pDq ´inftr P p0, `8q : ψ C,´r is not a diffeomorphism on its imageu • The radial system in pθ t , G t q tPr0, q :

@ t P r0, q, " d dt θ t " hpΨpG t , ? 2ζ t `θt qq @ x P BG t , B t x " α BGt, ? 2ζt`θt pxqν BGt pxq (50)
where R `Q t Þ Ñ ζ t P R is assumed to be α-Hölder regular with α P p0, 1{2q, is small enough and @ r P R, @ D P D, ΨpD, rq ď sPp´8,rs

ΨpC, sq

The interest of these manipulations is that a solution of (44) will be given by

@ t P r0, τq, D t ΨpG t , ? 2B t `θt q
where in (50) we take pζ t q tě0 " pB t q tě0 and where τ is the corresponding , which ends up being a stopping time with respect to the filtration generated by pB t q tě0 .

Local existence of a pushed mean curvature flow

Let F 0 : M Ñ R n`1 be a smooth immersion of an n-dimensional manifold M such that F 0 pM q " C. Let r : t P r0, 8q Þ Ñ rptq P R be a real continuous function. Consider the following equation, which is similar to (46) (i.e. BG t " F pt, M q), taking into account the remark made before Lemma 13:

" @ x P M, @ B Bt F pt, xq, ν F pt, xq D " ´ρb ΨpF pt,M q,rptqq pψ F pt,M q,rptq pxqq F p0, xq " F 0 pxq, (58)
where ν F pt, xq is the normal vector of the hypersurface F pt, M q at F pt, xq. The goal of this section is to show existence in small time of solution of (58) with enough regularity in space and time, under the hypotheses that rp0q is small enough and that r is α{2-Hölder regular, for some α P p0, 1q.

To get a small time existence of equation (58) we will convert the problem in terms of a quasiparabolic equation. We will study the linearisation of this equation, it turns to be linear and strictly parabolic for small time, with C α{2,α pr0, T s ˆM q coefficients when M is a C 2`α manifold. We will resort to an existing result on the existence and regularity of the solution of such a linear equation. Then we will use the inverse function theorem to get a solution of the original equation (58).

Let C " F 0 pM q, we will suppose that M is a C 3`α manifold and F 0 is a C 3`α diffeomorphism (in general we will denote by regpM q the manifold regularity of M ), so that C is also a C 3`α manifold. Small perturbations in time of C under (58) live in a small tubular neighborhood of C, and as in Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF], a useful way to obtain a quasi-linear equation from ( 58) is to represent the solution as graphs over the fixed hypersurface C. The underlying idea is to consider again the parametrization pr ´, r `q ˆC Q pr, yq Þ Ñ ψ C,r pyq of a tubular neighborhood of C, where pr ´, r `q is a small neighborhood of 0. Let x P M , and ν 0 pxq be the unit outward normal of the hypersurface C " F 0 pM q at the point F 0 pxq. Then one looks at the function f pt, .q : M Ñ R, with enough regularity, whose image is included into pr ´, r `q and which satisfies F pt, xq " ψ C,f pt,xq pF 0 pxqq " F 0 pxq `f pt, xqν 0 pxq, for all pt, xq P r0, S `q ˆM , with S `small enough, i.e. we represent F pt, M q as a graph over C, since C " F 0 pM q we have f p0, .q " 0 and the existence of S `is due to the regularity of f and the compactness of M .

Let x i be a local chart of M , g i,j p0, xq " xB i F 0 , B j F 0 y the Riemannian metric at x in this chart, g i,j p0, xq its inverse, h i,j p0, xq ΠpB i F 0 , B j F 0 q " x∇ B i F 0 pxq ν 0 pxq, B j F 0 pxqy where Π in the second fundamental form of C at F 0 pxq and define S i,j p0, xq " h i,k g k,l h lj p0, xq, where the convention that every repeated lower indices and upper indices is considered as a sum is enforced, as in the whole paper. We end up with the quasi-linear parabolic equation with respect to f in order that F pt, .q satisfies (58), after taking care that we have some dilation term rptq in the equation. We have for all i, j P n , t P r0, S `q

and x P M , B Bt F pt, xq " B t f pt, xqν 0 pxq B i ν 0 pxq " h i,k g k,l p0, xqB l F 0 pxq B i F pt, xq " B i F 0 pxq `f pt, xqh i,k g k,l p0, xqB l F 0 pxq `Bi f pt, xqν 0 pxq g i,j pt, xq " xB i F pt, xq, B j F pt, xqy " g i,j p0, xq `2f pt, xqh i,j p0, xq `f 2 pt, xqS i,j p0, xq `Bi f pt, xqB j f pt, xq G i,j pt, x, f, ∇f q νpt, xq " ν 0 pxq´B i f pt,xqg i,j pt,xqB j pF 0 pxq`f pt,xqν 0 pxqq }ν 0 pxq´B i f pt,xqg i,j pt,xqB j pF 0 pxq`f pt,xqν 0 pxqq} h i,j pt, xq " ´xνpt, xq, B i B j F pt, xqy " ´xνpt, xq, B i B j f pt, xqν 0 pxq `Bi B j F 0 pxq `Bi f pt, xqB j ν 0 pxq `Bj f pt, xqB i ν 0 pxq `f pt, xqB i B j ν 0 pxqy H i,j pt, x, f, ∇f, ∇∇f q (59)
where the second equality is the Gauss-Weingarten formula, where νpt, xq is the unit normal of the hypersurface F pt, M q at F pt, xq, and where we used the Gram-Schmidt procedure in the computation of νpt, xq (taking into account that `vi ? g i,l pt, xqB l F pt, xq ˘iP n is an orthonormal basis of T F pt,xq F pt, M q). To simplify the notations, denote G pG i,j pt, x, f, ∇f qq i,jP n and H pH i,j pt, x, f, ∇f, ∇∇f qq i,jP n , which take values in S nˆn , the space of symmetric matrices. Note that G does not depend on ∇∇f and that H has regularity regpM q ´3 in x (due to the term B i B j ν 0 pxq in H i,j pt, x, f, ∇f, ∇∇f q).

To manage the right hand side of (58), define

Ă M t ΨpF pt, M q, rptqq r F pt, xq Ψ F pt,M q,rptq pF pt, xqq " F pt, xq `rptqνpt, xq
and denote all the quantities that depend on Ă M t " r F pt, M q by the same letter as for F pt, M q with a tilde. So by the same computation as above we have for all i, j P n , t P r0, S `q and x P M :

B i r F pt, xq " B i F pt, xq `rptqB i νpt, xq " B i F pt, xq `rptqh i,k g k,l pt, xqB l F pt, xq r g i,j pt, xq " g i,j pt, xq `2rptqh i,j pt, xq `rptq 2 S i,j pt, xq " pGpId `2rptqG ´1H `rptq 2 G ´1H G ´1H qq i,j " ´G pId `rptqG ´1H q 2 ¯i,j r G i,j pt, x, f, ∇f, ∇∇f q r νpt, xq " νpt, xq r h i,j pt, xq " ´xνpt, xq, B i B j r F pt, xqy " H i,j ´rptqxνpt, xq, B i B j νpt, xqy " H i,j `rptqxB i νpt, xq, B j νpt, xqy " H i,j `rptqS i,j pt, xq " pH `rptqHG ´1H q i,j " pHpId `rptqG ´1H qq i,j r H i,j pt, x, ∇f, ∇∇f q (60)
As usual, denote r G p r G i,j pt, x, f, ∇f, ∇∇f q i,jP n , r H p r H i,j pt, x, f, ∇f, ∇∇f qq i,jP n and p r G i,j pt, x, f, ∇f, ∇∇f qq i,jP n r G ´1, all taking values in S nˆn , so that we have for the mean curvature ´ρΨpFpt,Mq,rptqq pψ F pt,M q,rptq pxqq " ´r G i,j r H i,j " ´trp r G ´1 r Hq

" ´tr ´pId `rptqG ´1H q ´2 G ´1H pId `rptqG ´1H q " ´tr ´pId `rptqG ´1H q ´1 G ´1H ¯ Φ1 pt, x, f, ∇f, ∇∇f q.
for some mapping Φ1 . Note that in the above formula only H depends on ∇∇f . Furthermore consider the mapping Φ2 such that

@ ∇U ´β, ν ΨpF pt,M q,rptqq D ψ F pt,M q,rptq pxq " A ∇U p r F pt, xqq ´βp r F pt, xqq, r νpt, xq E r F pt,xq " A ∇U p r F pt, xqq ´βp r F pt, xqq, νpt, xq E r F pt,xq Φ2 pt, x, f, ∇f q
Remark the above expression does not depend on ∇∇f . Define Φpt, x, f, ∇f, ∇∇f q Φ1 pt, x, f, ∇f, ∇∇f q `Φ 2 pt, x, f, ∇f q (61) so that Equation (58) becomes the following non-linear parabolic equation

$ & % B t f pt, xq " 1 xν 0 pxq,νpt,xqy Φpt, x, f, ∇f, ∇∇f q Φpt, x, f, ∇f, ∇∇f q f p0, xq " 0 (62)
Note that at time t " 0 we have f p0, xq " 0 , ∇f p0, xq " 0, ∇∇f p0, xq " 0. The application Φ defined above will be considered with the following argument Φpt, x, z, v, qq, where pt, xq P M T r0, T s ˆM , z P R, v P T x M and q is a symmetric matrix in T x M d T x M . Since r is continuous and rp0q " 0 (or small enough), for small T , Φ is smooth in three last variables in a neighborhood p0, 0, 0q and have at least the regularity regpM q ´3 in x, and the same Hölder regularity in time as r (i.e. it is enough to have G invertible and }rptqG ´1H } ă 1). More precisely we have the following proposition.

Proposition 19 There exist T ą 0 and R 0 ą 0 such that

• the mapping Φ : r0, T s ˆM ˆBp0 R ,0 R n ,0 S nˆn q pR 0 q Ñ R pt, x, z, v, qq Þ Ñ Φpt, x, z, v, qq (63)
is smooth in the three last components,

• the mapping t Þ Ñ Φpt, x, z, v, qq have the same Hölder regularity in time as r,

• the mapping x Þ Ñ Φpt, x, z, v, qq have at least the regularity regpM q ´3.

Proof

Recall that

Gpt, x, z, vq " Gp0, xq `2zHp0, xq `z2 Sp0, xq `v b v, νpt, x, z, vq " ν 0 pxq´v i g i,j pt,x,z,vqpB j F 0 pxq`zh j,k g k,l p0,xqB l F 0 pxq`v j ν 0 pxqq }ν 0 pxq´v i g i,j pt,x,z,vqpB j F 0 pxq`zh j,k g k,l p0,xqB l F 0 pxq`v j ν 0 pxqq} , H i,j pt, x, z, v, qq " ´xνpt, x, z, qq, ν 0 pxqyq i,j ´xνpt, x, z, qq,

B i B j F 0 pxqy ´xνpt, x, z, qq, v i B j ν 0 pxq `vj B i ν 0 pxq `zB i B j ν 0 pxqy (64)
Since Gp0, xq is invertible and M is compact, there exist R 0 , C 1 , C 2 ą 0 such that for |z|, }v}, }q} ď R 0 , Gpt, x, z, vq is invertible for all x P M,

}G ´1pt, x, z, vq} ď C 1 , }νpt, x, z, vq ´ν0 pxq} ď 1 2 , }Hpt, x, z, v, qq} ď C 2 .
Thus, since r is continuous and rp0q " 0 (or small enough), take T ą 0 such that

T sup # u ě 0 : sup sPr0,us |rpsq| ď 1 2C 1 C 2 + ( 65
)
Then }rptqG ´1H } ď 1 2 , and pId `rptqG ´1H q is invertible for all pt, x, z, v, qq P r0, T s ˆM Bp0 R ,0 R n ,0 S nˆn q pR 0 q, and the wanted conclusions easily follow.

Lemma 20 Let T be given by (65). For all pt, x, z, v, qq P r0, T s ˆM ˆBp0 R ,0 R n ,0 S nˆn q pR 0 q, we have:

B q i,j Φpt, x, z, v, qq " pG ´2rptqH `rptq 2 HG ´1H q ´1 i,j
Furthermore, B q Φpt, x, z, v, qq pB q i,j Φpt, x, z, v, qqq i,jP n is uniformly elliptic.

Proof

Let us write H as Hpqq Hpt, x, z, v, qq ´xνpt, x, z, vq, ν 0 pxqyq ´H1 pt, x, z, vq and recall that νpt, xq and G do not depend on ∇∇f , ie are constant in q. Consider ψpqq ´G´1 Hpqq, so Φpt, x, z, v, qq " tr `pId ´rptqψpqqq ´1 ψpqq ˘.

Let M P M nˆn , X P M nˆn small and u P R such that }upM `Xq} ă 1 then

pId ´upM `Xqq ´1pM `Xq " ř nPZ `pupM `Xqq n pM `Xq " ř nPZ `´u n M n`1 `un ř mP 0,n M m XM n´m ¯`opXq " pId ´uM q ´1M `řnPZ `, mP 0,n u n ř M m XM n´m `opXq (66)
so drpId ´uM q ´1M spXq " ř nPZ `, mP 0,n u n ř M m XM n´m . Hence d q ppId ´uψpqqq ´1qpX q " ÿ nPZ `, mP 0,n u n ψpqq m dψpqqpXqψpqq n´m Thus using the trace property d q trppId ´uψpqqq ´1ψpqqqpX q " tr ¨ÿ nPZ `, mP 0,n u n ψpqq m dψpqqpXqψpqq n´m ' " ÿ nPZ `, mP 0,n u n tr `ψpqq m dψpqqpXqψpqq n´m " ÿ nPZ `, mP 0,n u n tr pψpqq n dψpqqpXqq " tr ˜ÿ nPZ `pn `1qu n ψpqq n dψpqqpXq " tr `pId ´uψpqqq ´2dψpqqpX q ˘(67)

Thus we have d q Φpt, x, z, v, qqpXq " d q Φ1 pt, x, z, v, qqpXq " xνpt, x, z, vq, ν 0 pxqytrppId ´rptqG ´1H pqqq ´2G ´1X q so for any i, j P n ,

B q i,j Φpt, x, z, v, qq " xνpt, x, z, vq, ν 0 pxqyppId ´rptqG ´1H pqqq ´2G ´1X q j,i " xνpt, x, z, vq, ν 0 pxqypG ´2rptqH `rptq 2 HG ´1H q ´1 i,j
where G `2rptqH ´rptq 2 HG ´1H P S nˆn . For the last point of the lemma, use Proposition 19, and the choice of T in its proof, to get pG `2rptqH `rptq 2 HG ´1H q " GpId `rptqG ´1H q 2 is invertible for all t P r0, T s, and is continuous as function of t, so its spectrum remains positive as the spectrum at time 0, when rp0q " 0.

To show the existence result with sufficient regularity in time and space of Equation (58), we will show the existence result of the equivalent equation (62) up to a parametrization as in Proposition 1.3.4 in [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF]. We will intensively use the existence and regularity result of the linearised equation exposed in Lunardi [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF]. Let us recall briefly this result that appears as Theorem 5.1.10 of Lunardi [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF] and whose extension to the compact Riemannian manifold could be find e.g. as Theorem 2.3 of Huang [START_REF] Huang | The Cauchy problem for fully nonlinear parabolic systems on manifolds[END_REF] (with the bundle E " M ˆR).

For α P p0, 1q and T ą 0 let C α,0 pr0, T s ˆM q ! f P Cpr0, T s ˆM q : f p¨, xq P C α pr0, T sq, @x P M, and such that }f } C α,0 sup xPM t}f p¨, xq} C α pr0,T sq ă 8

)
where for any function

f : r0, T s Ñ R, }f } C α pr0,T sq }f } 8,r0,T s `xf y C α pr0,T sq (68)
xf y C α pr0,T sq sup " |f ptq ´f psq| |t ´s| α , s " t P r0, T s

* (69)
Similarly, we define C 0,α pr0, T s ˆM q ! f P Cpr0, T s ˆM q : f pt, .q P C α pM q, @t P r0, T s, and such that }f } C 0,α sup tPr0,T s t}f pt, ¨q} C α pM q ă 8

)
where the norm } ¨}C α pM q is defined as in (68) and (69), with r0, T s replaced by M . The most important functional spaces for our analysis will be, still for given 0 ă α ă 1, C α{2,α pr0, T s ˆM q C α{2,0 pr0, T s ˆM q X C 0,α pr0, T s ˆM q C 1`α{2,2`α pr0, T s ˆM q tf P C 1,2 pr0, T s ˆM q : B t f, B i B j f P C α{2,α pr0, T s ˆM q, @i, j P n u respectively endowed with the norms

}f } C α{2,α }f } C α{2,0 `}f } C 0,α }f } C 1`α{2,2`α }f } 8 `n ÿ i"1 }B i f } 8 `}B t f } C α{2,α `n ÿ i,j"1 }B i B j f } C α{2,α
As in Lemma 5.1.1 in Lunardi [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], there exists a uniform constant C α ą 0 such that for all f P C 1`α{2,2`α :

}B i f } C p1`αq{2,1`α ď C α }f } C 1`α{2,2`α (70)
Consider the following linear equation:

" B t f pt, xq " r g i,j pt, xqB i B j f pt, xq `ři r H 1,i pt, xqB i f pt, xq `r H 0 pt, xqf pt, xq `qpt, xq f p0, xq " f 0 pxq (71)
where r g pr g i,j q i,jP n , r H 1 p r H 1,i q iP n , r H 0 and q (respectively f 0 ) are some given mappings on M T r0, T s ˆM (resp. M ). As usual we will say that Equation ( 71) is uniformly elliptic in M T when there exists an ellipticity coefficient λ ą 0 such that for all t P r0, T s and all ξ 1 , ...., ξ n P R, we have:

r g i,j pt, xqξ i ξ j ě λ}ξ} 2 (72)
We recall the following theorem:

Theorem 21 (Th 5.1.10 Lunardi [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], Th 2.5 Huong [START_REF] Huang | The Cauchy problem for fully nonlinear parabolic systems on manifolds[END_REF]) Let r g, r H 1 , r H 0 and q belong to C α{2,α pr0, T s ˆM q, with 0 ă α ă 1 and let f p0, .q P C 2`α . Assume moreover that (71) is uniformly elliptic, i.e. (72) holds. Then there exists a quantity C ą 0, depending on the norms of r g, r H 1,i and r H 0 , as well as on the ellipticity coefficient of r g, such that Equation (71) has a unique solution f P C 1`α{2,2`α pr0, T s ˆM q and we have the Schauder estimate:

}f } C 1`α{2,2`α ď C `}f 0 } C 2`α `}q} C α{2,α
Let us come back to the original equation i.e. (62), we will consider the following space M t 0 " r0, t 0 s ˆM where the constant 0 ă t 0 ď T is to be chosen later, and let

X

tu P C 1`α{2,2`α pM t 0 q : up0, .q " 0, maxp}u} 8,Mt 0 , }∇u} 8,Mt 0 , }∇∇u} 8,Mt 0 q ď R 0 u

We define the map:

S : X Ñ C α{2,α pM t 0 q u Þ Ñ B t u ´Φpt, x, u, ∇u, ∇∇uq. (73)
This is clearly a continuously differentiable map.

We have the following theorem.

Theorem 22 Let M be a C 5`α manifold, for some fixed α P p0, 1q. If t Þ Ñ rptq is α{2-Hölder and rp0q " 0 then there exists t 0 ą 0 such that equation (62) has a unique solution defined in M t 0 with regularity C 1`α{2,2`α pM t 0 q.

Proof

The above theorem is a consequence of inverse function theorem around a specific function. Let u 0 pt, xq ş t 0 Φps, x, 0, 0, 0q ds and note that u 0 P C 1`α{2,2`α by the assumption on the regularity of M . The Fréchet derivative of S at u 0 it is given by

dSpu 0 qu " B t u ´`BΦ Bq i,j B i,j u `BΦ Bv i B i u `BΦ Bz u ˘,
where the coefficients are all evaluated at u 0 , for instance, BΦ Bq i,j stands for BΦ Bq i,j pt, x, u 0 , ∇u 0 , ∇∇u 0 q. By definition of u 0 , there exists 0 ă t 1 ď T such that for all 0 ď t ď t 1 , pu 0 , ∇u 0 , ∇∇u 0 qpt, xq P B p0 R ,0 R n ,0 S nˆn q pR 0 {2q, so u 0 P X. Lemma 20 yields BΦ Bq pt, x, u 0 , ∇u 0 , ∇∇u 0 q is strongly elliptic in M t 1 and is in C α{2,α pM t 1 q. Using Theorem 21, for the linearisation of (62), we get dSpu 0 q is locally invertible, and its inverse is continuous. By the inverse function theorem there exist ą 0, δ 1 ą 0 such that for all 0 ď t ď t 1 and for all g satisfying }g ´Spu 0 q} C α{2,α pMtq ă , there exists an unique f P C 1`α{2,2`α pM t q satisfying }f ´u0 } C 1`α{2,2`α pMtq ă δ 1 such that Spf q " g. For f such that }f ´u0 } C 1`α{2,2`α pMtq ă δ 1 , since f p0, xq " u 0 p0, xq " 0 and using (70), we get }f ´u0 } 8,Mt `}∇pf ´u0 q} 8,Mt `}∇∇pf ´u0 q} 8,Mt ď pt `Cα t pα`1q{2 `tα{2 qδ 1 (74)

where C α is the constant appearing in (70). So for t sufficiently small such that pt`C α t pα`1q{2 `tα{2 qδ 1 ď R 0 {2, we deduce f P X for 0 ă t 0 ď t.

Let us show that with respect to the C α{2,α pM t q norm, Spu 0 q tends to 0 as t goes to 0 `. We will first show that }Spu 0 qpt, xq ´Spu 0 qps, xq} ď C 1 pδq|t ´s| α{2 , for all s, t P r0, δs and x P M , and with C 1 pδq tending to 0 as δ tends to 0. We have, since M is compact and Φ is regular in the three last variables: with Cpδq tending to 0 as δ tends to 0. With the same computation as above, we have:

|d 3 Φpt, x, ζ σ pt, xqq ´d3
|Spu 0 qpt, xq ´Spu 0 qpt, yq| ď ż 1 0 | `d3 Φpt, x, ζ σ pt, xqq ´d3 Φpt, y, ζ σ pt, yqq ˘p u 0 pt, xqq| dσ `ż 1 0 | `d3 Φpt, y, ζ σ pt, yqq ˘p u 0 pt, xq ´ u 0 pt, yqq| dσ
We also have, since M is compact: |Φps, x, 0, 0, 0q ´Φps, y, 0, 0, 0q| ds ď Cδ|x ´y| α and in the same way, using the regularity of Φps, x, 0, 0, 0q in terms of x, we get:

|d 3 Φpt, x, ζ σ pt, xqq ´d3
| u 0 pt, xq ´ u 0 pt, yq| ď Cδ|x ´y| α
We deduce that:

|Spu 0 qpt, xq ´Spu 0 qpt, yq| ď Cpδq|x ´y| α
Hence }Spu 0 q} C α{2,α pMtq tends to 0 as t tends to 0. So there exist 0 ă t 2 such that }Spu 0 q} C α{2,α pMt 2 q ă . Let t 0 " t 1 ^t2 , we get by inverse function theorem that Sf " 0 has a solution f P C 1`α{2,2`α pM t 0 q, this is actually a solution of equation (62).

For the uniqueness, let f be the solution of (62) constructed above on M t 0 . Consider another solution g of (62) on M t 0 , in particular g starts with the same initial condition g 0 " f 0 " 0. Since g P C 1`α{2,2`α , let t 3 P p0, t 0 s be the maximum value of t such that }g} 8, Mt , }∇g} 8,Mt , }∇∇g} 8,Mt ď R 0 By construction of f , we have }f } 8,Mt , }∇f } 8,Mt , }∇∇f } 8,Mt ď R 0 for any t P r0, t 0 s and in particular for t P r0, t 3 s.

Let u " f ´g, then u satisfies the following linear equation:

B t u " Φpt, x, f, ∇f, ∇∇f q ´Φpt, x, g, ∇g, ∇∇gq " ż 1 0 B Bσ Φpt, x, σ f `p1 ´σq gq dσ " ż 1 0 BΦ Bq ij pt, x, σ f `p1 ´σq gqB i,j pf ´gq dσ `ż 1 0 BΦ Bv i pt, x, σ f `p1 ´σq gqB i pf ´gq dσ `ż 1 0 BΦ Bz pt, x, σ f `p1 ´σq gqpf ´gq dσ " A ij pt, xqu ij `Bi pt, xqu i `Cpt, xqu,
where

A ij pt, xq " ż 1 0 BΦ Bq ij pt, x, σ f `p1 ´σq gq dσ, B i pt, xq " ż 1 0 BΦ Bv i pt, x, σ f `p1 ´σq gq dσ, Cpt, xq " ż 1 0 BΦ Bz pt, x, σ f `p1 ´σq gq dσ.
According to Lemma 20, A ij is uniformly elliptic. Let λ ă ´}C} Mt 3 , and W e λt u then we have:

B t W " A i,j pt, xqB i B j W `Bi pt, xqW i `pC `λqW.
The proof of uniqueness will be done by contradiction, suppose f ‰ g then there exists for example β ą 0 (the negative possibility will be done in a similar way) and pt, xq P r0, t 3 s ˆM such that W pt, xq " β. Consider the first time t 0 such that there exist x 0 P M such that W pt 0 , x 0 q " β, clearly t 0 ą 0. By definition W pt 0 , x 0 q " maxtW pt, xq, pt, xq P r0, t 0 s ˆM u, and B t W pt 0 , x 0 q ě 0 HesspW qpt 0 , x 0 q ď 0 ∇W pt 0 , x 0 q " 0 We have at pt 0 , x 0 q

0 ď B t W " A ij pt 0 , x 0 qB i B j W `pC `λqβ ă A ij pt 0 , xqB i B j W ď 0
where the last inequality come from A ij pt 0 , x 0 qB i B j W " trpA HessW q ď 0, and this is a contradiction, so W ď 0. We do the same thing to get W ě 0 and so f " g for all t P r0, t 3 s. It follows in fact that t 3 " t 0 .

Remark 23 From the above proof, we see there exist two quantities η 1 , η 2 ą 0, only depending on some bounds on the geometry of C, such that t 0 can be expressed as

t 0 η 1 ^infts ě 0 : |rpsq| ě η 2 u
Remark 24 Using the α{2-Hölder regularity of the Brownian motion, for all 0 ă α ă 1, we get the existence and the regularity of the equation, similar to (44), corresponding to the stochastic modified mean curvature flow:

" D 0 " D @ t P r0, τq, @ x P C t , dx " `?2dB t ´ρb Ct pxqdt ˘νCt pxq (75)
where C t BD t . The solution of this equation is obtained as above, first we solve equation ( 58) and we obtain G t and then D t ΨpG t , ? 2B t q. Remark 25 Note that in the above proof we only need that rp0q is small enough, such that }rp0qG ´1H p0, ¨q} ă 1, so starting the same procedure at time t 0 , we have a notion of maximal solution of equation (62). A slight modification of the proof of Theorem 22 also yields existence and uniqueness of solution of (62) for f 0 small enough, as well as all its derivatives up to order 2.

Űsing the strong maximum principle instead of the maximum principle in the proof of Theorem 22, we have the following corollary:

Corollary 26 Let U, Û P D with C 5`α boundaries, α P p0, 1q, and C " BU, Ĉ " B Û . Suppose that

Û Ă U, Ĉ ‰ C
and that Ĉ belongs to an open tubular neighborhood of C. Let pBG t q tPr0,τ C q (resp. pB Ĝt q tPr0,τ Ĉ q ) be a solution of (58) with rptq " ? 2B t started at C (resp. Ĉ), then there exist a positive stopping time τ C, Ĉ ą 0 (a priori smaller than τ C ^τ Ĉ because we want BG t to remain in an open tubular neighborhood of B Ĝt ), such that @ t P p0, τ C, Ĉ q, Ĝt Ă G t , and BG t X B Ĝt " H

The above corollary shows that even if the initial hypersurfaces are equal in a large portion, it is sufficient they are different somewhere for the flow to detach them instantaneously, at least when one of them lives in a tubular neighborhood of the other. When the latter condition is not fulfilled, we have to impose that the initial boundaries are disjoint:

Corollary 27 Let U, Û P D with C 5`α boundaries, and C " BU, Ĉ " B Û . Suppose that Û Ă U, Ĉ X C " H Let pBG t q tPr0,τ C q (resp. pB Ĝt q tPr0,τ Ĉ q ) be a solution of (58) with rptq " ? 2B t started at C (resp. Ĉ), then for a positive stopping time τ C, Ĉ ą 0, we have @ t P r0, τ C, Ĉ q, Ĝt Ă G t , and BG t X B Ĝt " H Lemma 29 Let D, D P D with C 2 boundaries in a d-dimensional manifold V, C " BD, Ĉ " B D, D Ă D and Ĉ X C " H. Suppose that there exists k P R such that Ric ě pd ´1qkg, then at points pp, qq P C ˆĈ such that dpp, qq " dpC, Ĉq (or local minimizers of the distance function restricted to C ˆĈ) we have:

(i) if k ą 0,
(iii)
In particular for all k, if p is not conjugate to q then we have:

pd ´1qkdpp, qq ď ρ Ĉ pqq ´ρC ppq (iv) If V " R d then 0 ď ρ Ĉ pqq ´ρC ppq

Proof

Let pp, qq P C ˆĈ such that dpp, qq " dpC, Ĉq. Using the first variation formula, we get there exists an unit speed geodesic γ in V such that γp0q " q, γpdpp, qqq " p, 9 γp0q is orthogonal to T q p C and 9 γpdpp, qqq is orthogonal to T p C. Let pe i q iP 1,d´1 be a orthonormal basis of T q Ĉ. Let γ 1,i ptq be a geodesic in Ĉ such that γ 1,i p0q " q and 9 γ 1,i p0q " e i . Let γ 2,i ptq be a geodesic in C such that γ 2,i p0q " p and 9 γ 1,i p0q " {{ dpp,qq e i , where {{ is the parallel transport along the geodesic γ. We have 0 " xe i , 9

γp0qy " x{{ dpp,qq e i , 9 γpdpp, qqqy. Since pp, qq P C ˆĈ is a local minimizer of the distance function restricted to C ˆĈ, we have that

0 ď d 2 dt 2 ˇˇˇt "0 dpγ 1,i ptq, γ 2,i ptqq.
Let Y i be the Jacobi field along γ obtained by the variation of geodesic connecting γ 1,i ptq to γ 2,i ptq, we have: Y i p0q " e i , Y i pdpp, qqq " {{ dpp,qq e i . Using second variation formula, the fact that 9 γp0q is the exterior normal vector of Ĉ at q and 9 γpdpp, qqq is the exterior normal vector of C at p we get:

0 ď d 2 d 2 t dpγ 1,i ptq, γ 2,i ptqq| t"0 " " x∇ t"0 9 γ 2,i ptq, 9 γpdpp, qqqy ´x∇ t"0 9 γ 1,i ptq, 9 γp0qy ‰ `IpY i , Y i q " " x∇ t"0 9 γ 2,i ptq, ν C ppqy ´x∇ t"0 9 γ 1,i ptq, ν Ĉ pqqy ‰ `IpY i , Y i q " " ´x 9 γ 2,i p0q, ∇ 9 γ 2,i p0q ν C y `x 9 γ 1,i p0q, ∇ 9 γ 1,i p0q ν Ĉ y ‰ `IpY i , Y i q " ´ΠC
p{{ dpp,qq e i , {{ dpp,qq e i q `Π Ĉ pe i , e i q `IpY i , Y i q (77)

where IpY i , Y i q is the index of the Jacobi field Y i along γ, and Π C (resp. Π Ĉ ) is the second fundamental form of C (resp. Ĉ). Let X i psq " f psq{{ s e i , be a vector field along γ such that f p0q " f pdpp, qqq " 1 and f 2 " ´kf , using the Index Lemma since p and q are not conjugate along γ, we have for all i P 1, d ´1

IpY i , Y i q ď IpX i , X i q
Taking the sum in (77) we get:

0 ď d´1 ÿ i"1
`´Π C p{{ dpp,qq e i , {{ dpp,qq e i q `Π Ĉ pe i , e i q `IpY i , Y i q ď ρ Ĉ pqq ´ρC ppq `d´1 ż dpp,qq 0 pf 1 q 2 ´f 2 k ds " ρ Ĉ pqq ´ρC ppq `pd ´1qpf 1 pdpp, qqq ´f 1 p0qq.

ÿ i"1 IpX i , X i q " ρ Ĉ pqq ´ρC ppq `d´1 ÿ i"1 ż dpp,qq 0 }∇ s X i } 2 ´xRpX i ,
After computations of f , we get the result. For the particular case, we could take X i " {{ s e i in the above computation and directly get the result. If dpC, Ĉq ą 0, using Gauss Lemma, and the fact that D Ă D, we get the exterior normal vector of C at p is the parallel transport, along the geodesic γ that connects q to p, of the exterior normal vector of Ĉ at q. Hence by definition of Ψ we have dpΨ If dpp, qq " dpC, Ĉq " 0, since D Ă D then ν Ĉ pqq " ν C ppq and the result follows as above.

Remark 31

The above proposition also gives an alternative proof of Corollary 28. Let

ι V inf pp,vqPV ˆTpV : }v}"1 inftt ą 0, γ v ptq is conjugate to γ v p0q " pu
where γ v is a geodesic starting at γ v p0q " p and 9 γ v p0q " v.

Lemma 32 Let D, D P D with C 5`α boundaries, α P p0, 1q, and C " BD, Ĉ " B D. Suppose that there exists k ď 0 such that Ric ě pd ´1qkg, ι V " 8 (for example if V have non-positive sectional curvature) and

D Ă D, Ĉ X C " H.
Let pBD t q tPr0,τ C q (resp. pB Dt q tPr0,τ Ĉ q ) be a solution of (76) started at C (resp. Ĉ) then:

(i) The mapping t Þ Ñ dpBD t , B Dt q is locally Lipschitz in r0, τ C ^τ Ĉ q (ii) For all t P r0, τ C ^τ Ĉ q dpC, Ĉqe kpd´1qt ď dpBD t , B Dt q (iii) We have D t X Dt " H for all t P r0, τ C ^τ Ĉ q.

(iv) In particular, if V " R d then t Þ Ñ dpBD t , B Dt q is non decreasing in r0, τ C ^τ Ĉ q.

Proof

We have Recall that G t " tF 0 pxq `fC pt, xqν C 0 pxq, x P M u with F 0 pM q " C, and f C pt, xq the solution of (62). We have the same construction for Ĝ. We recall that f C P C 1`α{2,2`α pM τ C q and f Ĉ P C 1`α{2,2`α p Mτ Ĉ q. So by definition, for 0 ď t ă τ ,

D t "
dpBG t , B Ĝt q " inf px,yqPM ˆM dpF C pt, xq, F Ĉ pt, yqq
where F C pt, xq " F 0 pxq `fC pt, xqν C 0 pxq and F Ĉ pt, yq " F0 pyq `f Ĉ pt, yqν Ĉ 0 pyq. Also t Þ Ñ F C pt, xq and t Þ Ñ F Ĉ pt, yq are uniformly Lipschitz on any compact r0, T s Ă r0, τ q. Hence t Þ Ñ dpBG t , B Ĝt q " dpBD t , B Dt qq is Lipschitz on r0, T s, hence almost everywhere differentiable on r0, T s and absolutely continuous. At differentiability time t P r0, T s we have ´ρBDt px t q `ρB Dt py t q ě pd ´1qkdpBD t , B Dt q where in the second equality we use the usual Lagrange Theorem, in the third one we use the first variation formula, and in the last one we use Lemma 29. Since t Þ Ñ dpBD t , B Dt q is absolutely continuous we can integrate the above inequality. Hence, using Gronwall's lemma, we get the conclusions (i), (ii), (iii) and (iv) of the lemma, at least on r0, τ q. Since dpC, Ĉq ą 0, we easily deduce that τ " τ C ^τ Ĉ .

d
Remark 33 If the D Ă D c and C X Ĉ " H for all reasonable r, we have dpΨ C pp, rq, Ψ Ĉ pq, rqq " dpp, qq ´2r and we could get a similar kind of result. Theorem 34 Let D, D P D with C 5`α boundaries, α P p0, 1q, and C " BD, Ĉ " B D. Suppose that there exists k P R such that Ric ě pd ´1qkg and ι V ą 0 (for example if the sectional curvature is bounded above by a 2 then ι V ě π a , see e.g. [START_REF] Gallot | Riemannian geometry. Universitext[END_REF] page 159) and D Ă D, Ĉ X C " H Let pBD t q tPq0,τ C q (resp. pB Dt q tPr0,τ Ĉ q ) be a solution of (76) started at C (resp. Ĉ) then (i) The mapping t Þ Ñ dpBD t , B Dt q is locally Lipschitz on r0, τ C ^τ Ĉ q.

(ii) If k ě 0 then for all t P r0, τ C ^τ Ĉ q, pdpC, Ĉqe kpd´1qt q ^ιV ď dpBD t , B Dt q (iii) If k ď 0 then for all t P r0, τ C ^τ Ĉ q, pdpC, Ĉq ^ιV qe kpd´1qt ď dpBD t , B Dt q (iv) We have D t X Dt " H for t P r0, τ C ^τ Ĉ q.

Proof

The proof is similar to the proof of Lemma 32. Using (iii) in Lemma 29, we have:

dpBD t , B Dt q ă ι V ùñ d dt dpBD t , B Dt q ě pd ´1qkdpBD t , B Dt q
We deduce that, if k ě 0 then for all t P r0, τ C ^τ Ĉ q pdpC, Ĉqe kpd´1qt q ^ιV ď dpBD t , B Dt q since after being above ι V , dpBD t , B Dt q cannot go below ι V again. Similarly, if k ď 0 then for all t P r0, τ C ^τ Ĉ q pdpC, Ĉq ^ιV qe kpd´1qt ď dpBD t , B Dt q

As a consequence of Theorem 34, we can extend Corollary 26 under an assumption relaxing the requirement that one of the initial boundaries must be in a tubular neighborhood of the other initial boundary:

Proposition 35 Let D, D P D with C 5`α boundaries, α P p0, 1q, and C " BD, Ĉ " B D. Suppose that D Ă D and C ‰ Ĉ Let pBD t q tPr0,τ C q (resp. pB Dt q tPr0,τ Ĉ q ) be a solution of (76) started at C (resp. Ĉ). Suppose that there exists k P R such that Ric ě pd ´1qkg, ι V ą 0, and (H): it is possible to interpolate between C and Ĉ by a family of C 5`α hypersurfaces pC i q iP 0,n such that C i " BD i with D i P D, C i is in a tubular neighborhood of C i`1 , and

D Ă D i`1 Ł D i Ă D, for i P 0, n ´1 , C 0 " C and C n " Ĉ. Then (i) The mapping t Þ Ñ dpBD t , B Dt q is locally Lipschitz on r0, τ C ^τ Ĉ q.
(ii) BD t X B Dt " H, for t P p0, τ C ^τ Ĉ q.

Proof

We can use Corollary 26 with initial conditions C i and C i`1 , and extend this corollary without the hypothesis that Ĉ belongs to an open tubular neighborhood of C, up to the time τ

C, Ĉ inf iP 1,n´1 τ C i ,C i`1 .
Hence for all t P p0, τ C, Ĉ q and all i P 1, n ´1 we have pG i`1 q t Ă pG i q t and BpG i`1 q t X BpG i q t " H so for all t P p0, τ Using the same reasoning as the proof of Theorem 34, since BD t " ΨpBG t , ? 2B t q and B Dt " ΨpB Ĝt , ? 2B t q for all t P r0, τ q, we get @t P r0, τ q, dpBD t , B Dt qq " dpBG t , B Ĝt q and t Þ Ñ dpBD t , B Dt q is locally Lipschitz on r0, τ q Hence using (78) we get @ t P p0, τ q, Dt Ă D t and BD t X B Dt " H

Let t 0 " τ C, Ĉ 2
, since Dt 0 Ă D t 0 and dpBD t 0 , B Dt 0 q ą 0 we apply (ii) or (iii) of Theorem 34 to Dt 0 Ă D t 0 . We get, independently of the sign of the constant k, @ t P rt 0 , τ Ct 0 ^τ Ĉt 0 q, dpBD t , B Dt q ą 0 since τ Ct 0 " τ C ´t0 and τ Ĉt 0 " τ Ĉ ´t0 we have τ " τ C ^τ Ĉ .

Remark 36 In the above proposition, Hypothesis (H) seems to be satisfied for all D, D P D with D Ă D, even if BD X B D ‰ H, but for the moment we do not have a complete proof of this fact.

4.2

Local existence of (51)

In this subsection, we will show the existence of a solution to the system of equations (51). As the basic principle described in the paragraph following [START_REF] Machida | Λ-linked coupling for drifting Brownian motions[END_REF], a solution of (51) could be obtained as a solution of (58) conditioned not to collapse. Unfortunately, to develop this approach, we would need a solution of (58) defined for all times up to this collapsing. Since we have not been able to find such a maximal solution, we will directly work on (51), inspired by the previous subsection. We recall the notations: @ D P D, C " BD hpDq " 2 µpCq µpDq @ r ą 0, @ D P D r , @ x P C, α C,r pxq ´ρb ΨpC,rq pψ C,r pxqq.

For given D 0 P D, we are interested in the system of equations:

$ & % d dt θ t " hpΨpG t , ? 2B t `θt qq @ x P BG t , B t x " α BGt, ? 2Bt`θt pxqν BGt pxq pθ 0 , G 0 q " p0, D 0 q (79)
To prove the existence of a solution to the above system of equations, we consider the equation described below. Let g : r0, `8q Q t Þ Ñ gptq P R be a real α 2 -Hölder function, such that gp0q " 0 (or small enough), and 0 ă α ă 1. The goal of this first step is to show the existence of real numbers t 0 ą 0 and δ ą 0, such that for all g P B C α{2 p0, δq and gp0q " 0 , there exists a family pG g t q tPr0,t 0 s solution of " @ t P r0, t 0 s, @ x P BG g t , B t x " α BG g t , ?

2Bt`gptq pxqν BG g t pxq G g 0 " D 0 (80)

We adopt the same strategy as in the previous section, in order to deal with the quasi-parabolic equation, and we adopt the same notation, let BD 0 " F 0 pM q.

We consider the following equation.

" @ B Bt F g pt, xq, ν F g pt, xq D " ´ρb ΨpF g pt,M q, ? 2Bt`gptqq pψ F g pt,M q, ? 2Bt`gptq pxqq F p0, xq " F 0 pxq, (81)
As before we represent the solution as graphs over the fixed hypersurface C " F 0 pM q, and we write the solution as: F g pt, xq " ψ C,f g pt,xq pF 0 pxqq " F 0 pxq `f g pt, xqν 0 pxq for a function f g with enough regularity and f g p0, .q " 0. With similar computations as in the above section, F g is a solution of (81) (with rptq " ? 2B t `gptq for any t ě 0) if f g satisfy the following non linear parabolic equation:

" B t f g pt, xq " Φ g pt, x, f g , ∇f g , ∇∇f g q f g p0, xq " 0, ( 82
)
where Φ g have the same definition as Φ in Proposition 19, but with rptq " ? 2B t `gptq, for all t ě 0. Taking into account that C is smooth, Theorem 22 leads to:

Proposition 37 Take g " g 0 " 0. There exists 0 ă t 0 ď T (where T comes from Proposition 19) such that (82) admits a solution f g 0 belonging to Xpt 0 q tu P C 1`α{2,2`α pM t 0 q : up0, .q " 0, maxp}u} 8,Mt 0 , }∇u} 8,Mt 0 , }∇∇u} 8,Mt 0 q ď R 0 u

We deduce:

Proposition 38 With the same notation as in the above proposition, there exist two real δ 0 , δ 1 ą 0 and a continuously differentiable map

Θ : B C α 2 pr0,t 0 sq pg 0 , δ 0 q Ñ B C 1`α 2 ,2`α pMt 0 q pf g 0 , δ 1 q g Þ Ñ f g , (83)
where f g is a solution of (82). Moreover Θ is uniformly Lipschitz in B C α 2 pr0,t 0 sq pg 0 , δ 0 q.

Proof

Consider the mapping

S : Xpt 0 q ˆC α 2 pr0, t 0 sq Ñ C α 2 ,α pM t 0 q pu, gq Þ Ñ B t u ´Φg pt, x, u, ∇u, ∇∇uq (84)
It is continuously differentiable, at least when g belongs to a small ball. Note that from Proposition 37, there exists pf g 0 , g 0 q P Xpt 0 q ˆC α 2 pr0, t 0 sq such that Spf g 0 , g 0 q " 0. Also dS u pf g 0 , g 0 qpvq " dSpf g 0 qpvq where S is defined before the proof of Theorem 22 (with rptq ? 2B t ). Since f g 0 is in X, dSpf g 0 q is invertible with continuous inverse, according to Lemma 20 and Theorem 21. The result follows from implicit function theorem.

We will show the existence of solution of (79) by using a fixed point theorem. For g P B C α 2 pr0,t 0 sq pg 0 , δ 0 q, define F g pt, xq F 0 pxq `f g pt, xqν 0 pxq and consider the family of hypersurfaces

BG g t F g pt, M q note that G g 0 " D 0 .
Proposition 39 There exist 0 ă t 1 ď t 0 and a mapping

Γ : B C α 2 pr0,t 1 sq pg 0 , δ 0 q X tg P C α 2 |gp0q " 0u Ñ B C α 2 pr0,t 1 sq pg 0 , δ 0 q X tg P C α 2 : gp0q " 0u such that " @ t P r0, t 1 s, d dt Γpgqptq " hpΨpG g t , ? 2B t `gptqqq, Γpgqp0q " 0. ( 85
)
Moreover Γ is a contraction and there exists an unique fixed point for Γ in B C α 2 pr0,t 1 sq pg 0 , δ 0 q X tg P C α 2 pr0, t 1 sq : gp0q " 0u.

Proof

Take δ 0 such that by Proposition 38, Θ is uniformly Lipschitz in B C α 2 pr0,t 0 sq pg 0 , δ 0 q. Let g P B C α 2 pr0,t 0 sq pg 0 , δ 0 q, r P R and f g " Θpgq, define for all x P M :

F g ψ pt, x, rq " F g pt, xq `rν F g pt, xq " F 0 pxq `f g pt, xqν 0 pxq `rν F g pt, xq " F 0 pxq `f g pt, xqν 0 pxq `rνpt, x, f g pt,
? 2B t `gptqqq " 1 n`1 ş F g ψ pt,M q x x, ν F g ψ y dµ F g ψ pt,M q p xq " 1 n`1 ş M xF g ψ pt, x, ? 2B t `gptqq, ν F g ψ pt, xqyd g pt, xq dµ M pxq. (88)
As before, we get, for some C 1 δ 1 ,δ 0 ą 0 and K 1 ą 0 of the same nature as C δ 1 ,δ 0 ą 0 and K ą 0, that }t Þ Ñ µpΨpG g t ,

? 2B t `gptqqq} C α{2 ď pC 1 δ 1 ,δ 0 pt α{2 0 `1q `K1 qµpM q (89)
As a quotient, it follows that t Þ Ñ hpΨpG g t , |t ´s| α{2 , since Γpgqp0q " 0 we have that:

? 2B t `gptqq is in C α{2 ,
}Γpgq} C α{2 r0,t 1 s ď Ct 1 `Ct 1´α{2 1
.

Take 0 ă t 1 ď t 0 sufficiently small such that Ct 1 `Ct

1´α{2 1 ď δ 0 we have Γ maps B C α 2 pr0,t 1 sq pg 0 , δ 0 q X tg P C α 2 |gp0q " 0u into himself.
Let us show that Γ is a contraction. Let g 1 , g 2 P B C α 2 pr0,t 1 sq pg 0 , δ 0 q, and f g 1 " Θpg 1 q, f g 2 " Θpg 2 q then µpΨpBG g

2}B.} C α{2 `2δ 0 `2δ 1 q ď C δ 0 ,δ 1 |t ´s| α{2
Also using the regularity of V in the four last variables we have Jpt, xq "

ż 1 0 d 3 V px, ζ σ pt, xqqppg 1 ptq, f g 1 pt, xqq ´pg 2 ptq, f g 2 pt, xqqq dσ Hence, |Jpt, xq ´Jps, xq| " | ż 1 0 d 3 V px, ζ σ pt, xqqpg 1 ptq ´g2 ptq, f g 1 pt, xq ´ f g 2 pt, xqq ´d3 V px, ζ σ ps, xqqpg 1 psq ´g2 psq, f g 1 ps, xq ´ f g 2 ps, xqqdσ| ď ż 1 0 | `d3 V px, ζ σ pt, xqq ´d3 V px, ζ σ ps, xqq ˘pg 1 ptq ´g2 ptq, f g 1 pt, xq ´ f g 2 pt, xqq| dσ `ż 1 0 ˇˇd 3 V px, ζ σ ps, xqq `pg 1 ptq ´g2 ptq, f g 1 pt, xq ´ f g 2 pt, xqq ´pg 1 psq ´g2 psq, f g 1 ps, xq ´ f g 2 ps, xqq ˘ˇd σ Since d 3 V px, ζ σ ps,
xqq is bounded we have (again the constant C can change from one line to the other),

|d 3 V px, ζ σ ps, xqq `pg 1 ptq ´g2 ptq, f g 1 pt, xq ´ f g 2 pt, xqq ´pg 1 psq ´g2 psq, f g 1 ps, xq ´ f g 2 ps, xqq ď C|t ´s| α{2 p}g 1 ´g2 } C α{2 `} f g 1 ´ f g 2 } C α{2,α q ď C|t ´s| α{2 p}g 1 ´g2 } C α{2 `}Θpg 1 q ´Θpg 2 q} C 1`α{2,2`α q ď C|t ´s| α{2 p1 `}Θ} Lip q}g 1 ´g2 } C α{2
where in the last line we use Proposition 38. Using that d 3 V px, .q is Lipschitz in the last variable:

|d 3 V px, ζ σ pt, xqq ´d3 V px, ζ σ ps, xqq ď C|ζ σ pt, xq ´ζσ ps, xq| ď CC δ 0 ,δ 1 |t ´s| α{2
Since pg 1 p0q, f g 1 p0, xqq " 0 " pg 2 p0q, f g 2 p0, xqq we have:

|pg 1 ptq ´g2 ptq, f g 1 pt, xq ´ f g 2 pt, xqq| ď Ct α{2 p1 `}Θ} Lip q}g 1 ´g2 } C α{2
Putting all things together we get xt Þ Ñ Jpt, xqy C α{2 ď C}g 1 ´g2 } C α{2 and since Jp0, xq " 0,

}t Þ Ñ Jpt, xq} C α{2 ď C}g 1 ´g2 } C α{2 Hence }t Þ Ñ µpΨpBG g 1 t , ? 2B t `g1 ptqqq ´µpΨpBG g 2 t , ? 2B t `g2 ptqqq} C α{2 ď C}g 1 ´g2 } C α{2 (92)
With the same proof we also have:

}t Þ Ñ µpΨpG g 1 t , ? 2B t `g1 ptqqq ´µpΨpG g 2 t , ? 2B t `g2 ptqqq} C α{2 ď C}g 1 ´g2 } C α{2 (93)
Let µpgqptq µpΨpG g t , ?

2B t `gptqqq and µpgqptq µpΨpBG g t , ?

2B t `gptqqq d dt ´Γpg 1 q ´Γpg 2 q ¯" 2 ´µpg 1 q µpg 1 q ´µpg 2 q µpg 2 q " 2 ´µpg 1 qµpg 2 q ´µpg 2 qµpg 1 q µpg 1 qµpg 2 q " 2 ´µpg 1 qpµpg 2 q ´µpg 1 qq ´µpg 1 qpµpg 2 q ´µpg 1 qq µpg 1 qµpg 2 q Hence using (87), ( 89), ( 92) and (93),

› › › › d dt ´Γpg 1 q ´Γpg 2 q ¯› › › › C α{2 pr0,t 1 sq ď C}g 1 ´g2 } C α{2
and so

› › › › d dt ´Γpg 1 q ´Γpg 2 q ¯› › › › C 0 pr0,t 1 sq ď C}g 1 ´g2 } C α{2 Since Γpg 1 qp0q " 0 " Γpg 2 qp0q, }Γpg 1 q ´Γpg 2 q} C α{2 pr0,t 1 sq ď pt 1 `t1´α{2 1 qC}g 1 ´g2 } C α{2
Reducing t 1 such that pt 1 `t1´α{2

1 qC ď 1 2 , we get:

}Γpg 1 q ´Γpg 2 q} C α{2 pr0,t 1 sq ď 1 2 }g 1 ´g2 } C α{2 pr0,t 1 sq
Hence Γ have a unique fixed point in B C α 2 pr0,t 1 sq pg 0 , δ 0 q X tg P C α 2 |gp0q " 0u.

Theorem 40 Let D 0 P D, then there exists 0 ă t 1 such that the system of equations (79) has a unique solution.

Proof

Let θ be the fixed point of Γ, and f θ " Θpθq then F θ pt, xq " F 0 pxq `f θ pt, xqν 0 pxq solves

# B Bt F θ pt, xq " `´ρ b ΨpF θ pt,M q, ?
2Bt`θptqq pψ F θ pt,M q, ? 2Bt`θptq pxqq ˘νF θ pt, xq F p0, xq " F 0 pxq. and so

" @ t P r0, t 1 s, @ x P BG θ t , B t x " α BG θ t , ? 2Bt`θptq pxqν BG θ t pxq G θ 0 " D 0 Also $ & % d dt θptq " d dt Γpθqptq " hpΨpG θ t , ? 2B t `θptqq, Γpθqp0q " 0.
Let D P D, C " BD with C 5`α boundaries, α P p0, 1q, in a d-dimensional Riemannian manifold V , and pθ t , G t q 0ďtăτ be a solution of (79) given by Theorem 40. As in the beginning of this section, the solution of @ t P r0, τq, @ x P C t D t , dx " ˆ?2dB t `2 µpC t q µpD t q dt ´ρCt pxqdt ˙νCt pxq (94) is given by pD t q tPr0,τq , where

@ t P r0, τq, D t ΨpG t , ? 2B t `θt q
(as a special case of (44)). Proposition 42 below will give a control of the extrinsic diameter of C t defined by

diampC t q sup px,yqPC 2 t dpx, yq
where dp¨, ¨q is the Riemannian distance in V . First we need the following proposition bounding the sum of the mean curvature at points that realize the diameter, in terms of the extrinsic curvature (by extrinsic we mean in the ambient manifold V , i.e. not intrinsic in the hypersurface). For all b P R, we denote by V b pdq the d-dimensional manifold with constant curvature b. Let ι V b pdq defined before Lemma 32. We have:

" ι V b pdq " 8 , if b ď 0 ι V b pdq " π ? b
, if b ą 0

Proposition 41 Let D P D with a C 2 boundary in a d-dimensional manifold V , and C " BD. Suppose that there exists b P R such that the sectional curvature K V of V is bounded above by b, i.e. K V ď b. For all pp, qq P C 2 such that dpp, qq " diampCq and dpp, qq ă ι V b pdq , we have ˘.

Proof

As in the proof of Lemme 29, consider pp, qq P C 2 such that dpp, qq " diampCq. Using the first variation formula, we get there exists an unit speed geodesic γ in V such that γp0q " q, γpdpp, qqq " p, 9 γp0q " ´νC pqq and 9 γpdpp, qqq " ν C ppq. Let pe i q iP 1,d´1 be a orthonormal basis of T q C. For i P 1, d ´1 , let γ 1,i ptq be a geodesic in C such that γ 1,i p0q " q and 9 γ 1,i p0q " e i . Let γ 2,i ptq be a geodesic in C such that γ 2,i p0q " p and 9 γ 1,i p0q " {{ dpp,qq e i , where {{ is the parallel transport along the geodesic γ. Since pp, qq P C 2 is a local maximum of the distance function restricted to C ˆC, we have that

d 2 dt 2 ˇˇˇt "0 dpγ 1,i ptq, γ 2,i ptqq ď 0.
Let Y i be the Jacobi field along γ obtained by the variation of geodesic connecting γ 1,i ptq to γ 2,i ptq, we have: Y i p0q " e i , Y i pdpp, qqq " {{ dpp,qq e i . Using second variation formula, we get:

d 2 d 2 t dpγ 1,i ptq, γ 2,i ptqq| t"0 " " x∇ t"0 9 γ 2,i ptq, 9 γpdpp, qqqy ´x∇ t"0 9 γ 1,i ptq, 9 γp0qy ‰ `IpY i , Y i q " " x∇ t"0 9 γ 2,i ptq, ν C ppqy ´x∇ t"0 9 γ 1,i ptq, ´νC pqqy ‰ `IpY i , Y i q " ´ΠC
p{{ dpp,qq e i , {{ dpp,qq e i q ´ΠC pe i , e i q `IpY i , Y i q Put the above two computations together and take the sum to get:

´ρC pqq ´ρC ppq ď ´d´1 ÿ i"1
IpY i , Y i q.

We have to bound from below the index of the normal Jacobi field Y i for all i. Since Y i is a normal Jacobi field, there exist real functions f j i for j P 1, d ´1 such that Y i ptq " ř d´1 j"1 f j i ptq{{ t e j . By construction of Y i , we have f j i p0q " f j i pdpp, qqq " δ j i . Consider γptq tPr0,dpp,qqs a geodesic in V b pdq with same length as γ, take pẽ i q iP 1,d´1 an orthonormal basis of 9 γp0q K in T γp0q V b pdq, and denote by {{ the parallel transport along γ. Let Xi ptq " ř d´1 j"1 f j i ptq {{ t ẽj , be a vector field along γ, note that Xi p0q " ẽi and Xi pdpp, qqq " {{ẽ i . Let Ỹi be the Jacobi field in V b pdq along γ such that Ỹi p0q " ẽi and Ỹi pdpp, qqq " {{ dpp,qq ẽi . We have by definition:

IpY i , Y i q " ż dpp,qq 0 }∇ t Y i } 2 ´xRpY i , 9 γqY i , 9 γy dt ě ż dpp,qq 0 }∇ t Y i } 2 ´b}Y i } 2 dt " ż dpp,qq 0 }∇ t Xi } 2 ´b} Xi } 2 dt ě ż dpp,qq 0 }∇ t Ỹi } 2 ´b} Ỹi } 2 dt
where in the last inequality we used again the Index Lemma, since dpp, qq ă ι V b pdq . So γp0q and γpdpp, qqq are not conjugate in V b pdq. Since Ỹi ptq " f b ptq {{ t ẽi with f 2 b " ´bf b , and f b p0q " f b pdpp, qqq " 1, we get

IpY i , Y i q ě ż dpp,qq 0 pf 1 b q 2 ´bf 2 b dt " pf 1 b pdpp, qqq ´f 1 b p0qq
Hence ´ρC pqq ´ρC ppq ď ´pd ´1qpf 1 b pdpp, qqq ´f 1 b p0qq

and the result follows by explicit computation of f b in different cases.

Proposition 42 Let D P D with a C 5`α boundary C BD in a d-dimensional manifold V , for some fixed α P p0, 1q. Suppose there exists b P R such that the sectional curvature of V satisfies K V ď b. Then the evolution of the diameter of the solution pC t q tPr0,τq of (94) started at C is controlled by: (i) If b ď 0, we get for all 0 ď t ă τ, pdiampC . qq inftt ě 0 : diampC t q ě π ? b u.

d diampC t q ď 2p ? 2dB t `

Proof

Using the construction of pD t q tPr0,τq , we get, for 0 ď t ă τ,

diampC t q " sup px,yqPBG 2 t dpΨ Gt px, ? 2B t `θt q, Ψ Gt py, ? 2B t `θt qq " 2p ? 2B t `θt q `sup px,yqPBG 2 t dpx, yq
where in the second equality, we used that for 0 ď t ă τ, Ψ Gt p., ? 2B t `θt q is a diffeomorphism onto its image, and a reasoning similar to the proof of Proposition 30. Also since sup px,yqPBG ´ρBDt px t q ´ρBDt py t q

Taking into account Proposition 41, we obtain the wanted points (i) and (ii).

When ( 94) is replaced by (76), the previous proof leads to a similar result:

Proposition 43 Let D P D with a C 5`α boundary C BD in a d-dimensional manifold V , for some fixed α P p0, 1q. Suppose there exists b P R such that the sectional curvature of V satisfies K V ď b. Then the evolution of the diameter of the solution pC t q tPr0,τq of (76) started at C is controlled by: (i) If b ď 0, we get, for all 0 ď t ă τ, pdiampC . qq inftt ě 0 : diampC t q ě π ? b u.

d
Remark 44 Proposition 43 may seem simpler than Proposition 42, since it does not require to deal with the tricky term hpD t q. For instance when K V ď 0, we have for all 0 ď t ă τ:

diampC t q ď 2 ? 2pB t ´B0 q `diampC 0 q
It follows that τ ď τ ´diampC 0 q 5 Back to the homogeneous situations

Here we return to the situations encountered in Section 2, where V has a constant curvature and is endowed with the Laplacian L ∆. This section has two main goals developed in the following subsections:

• When V is an Euclidean space, it is possible to go further in the considerations of Section 3. In particular when V " R 2 , it is possible to compute explicitly the image of the mean curvature vector field by the tangent mappings to the normal flow.

• When D 0 " Bpx 0 , r 0 q with x 0 P V and r 0 ą 0 (small enough in the spherical case), the Doss-Sussman approach can be described explicitly (more generally this is also true when V is rotationally symmetric and x 0 is a center of symmetry). It is then possible to compare the Doss-Sussman methods in the two decompositions ( 22) and (57), concerning their respective time-domains and to see that the method suggested in Remark 18 is stable when we let r 0 go to zero, namely when we try an approximation of the initial conditions consisting of singletons.

About the Euclidean and constant curvature spaces

We begin by bringing some precisions about the quantities defined in ( 25) and [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF]. They can always be written

R ´pDq " r R ´pDq _ p R ´pDq and R `pDq " r R `pDq ^p R `pDq
where r R `pDq inftr P p0, `8q : ψ C,r is not an immersionu r R ´pDq ´inftr P p0, `8q : ψ C,´r is not an immersionu p R `pDq inftr P p0, `8q : ψ C,r is not one-to-oneu p R ´pDq ´inftr P p0, `8q : ψ C,´r is not one-to-oneu (with the usual convention inf H " `8).

Consider the Euclidean case:

Lemma 45 When V " R n , with n ě 2 and endowed with its Euclidean structure, we have

r R ´pDq " 1 minp0 ´, mint´λ n´1,C pxq : x P Cuq P r´8, 0q r R `pDq " 1 maxp0 `, maxt´λ 1,C pxq : x P Cuq P p0, `8s
where λ 1,C pxq ď ¨¨¨ď λ n´1,C pxq are the eigenvalues of the second fundamental form (defined with respect to ν C ) at x P C. The notations 0 ´and 0 `just indicate that 1{0

´" ´8 and 1{0 `" `8.

Proof

Recall that the tangent mapping dν C associated to the mapping C Q x Þ Ñ ν C pxq can be seen as a linear mapping from T x C (the tangent space of C at x) to itself, and that the second fundamental form is given at x P C by

T x C ˆTx C Q pv, wq Þ Ñ xv, dν C rwsy
We deduce that for r P R, the tangent mapping dψ C,r satisfies @ v, w P T x C, xv, dψ C,r rwsy " xv, wy `r xv, dν C rwsy It follows that if r is such that all the quantities 1 `rλ C,1 pxq, ..., 1 `rλ C,n´1 pxq are either all positive or all negative, then the tangent mapping dψ C,r is not degenerate at x. As a consequence, for r P p r R ´pDq, r R `pDqq, dψ C,r is not degenerate on C. More precisely, p r R ´pDq, r R `pDqq is the largest interval I containing 0 on which the tangent mapping dψ C,r is not degenerate on C for all r P I. Indeed, when for some x P C and r P R, the values 1 `rλ C,1 pxq, ..., 1 `rλ C,n´1 pxq are not of the same sign, we can find r 1 P p´|r|, |r|q such that 1 `r1 λ C,1 pxq " 0, so that dψ C,r 1 is degenerate at x.

Remark 46 Consider the case where V " R 2 endowed with its usual Riemannian structure (coming from its Euclidean structure). The following picture (where the boundary of the C in black stands for C, while the line in red is a portion of its image by ψ C,r , for some positive element r P p r R ´pDq, r R `pDqq), shows that in general the mapping ψ C,r is not an embedding of C in the plane. In this picture, if r is reduced a little to be equal to p R `pDq and if x " x 1 P C are such that ψ C,r pxq " ψ C,r px 1 q, it appears that ν C pxq " ´νC px 1 q and x 1 belongs to the line passing by x and directed by ν C pxq.

The last observation of the above remark corresponds to a general phenomenon that we now describe, coming back to the situation of an abstract Riemannian manifold V . The interest of these quantities is:

Lemma 47 When p R `pDq ă r R `pDq, it means that R `pDq " p R `pDq " Ȓ`p Dq ą 0. Similarly, we always have R ´pDq " r R ´pDq _ Ȓ´p Dq ă 0.

Proof

We only prove the first assertion, since the second one can be shown in the same way, by reversing the time (or, when V is compact, by replacing D by D c ). We begin by remarking that for any x P C, we can find a neighborhood U of x such that the intersection of U X C and U X exp x pr´ , sν C pxqq is reduced to x for ą 0 small enough (this is a consequence of the assumption that C is a smooth submanifold of V ). It follows that the set tr ą 0 : exp x prν C pxqq P C and ν C pexp x prν C pxqqq " ´d dr exp x prν C pxqqu does not contain 0 as an accumulation point. Since it is also closed, for any x P C, the infimum defining Ȓ`p xq is either attained and positive or infinite. Assume that Ȓ`p Dq ă `8 and let px n q nPN be a sequence of elements of C such that Ȓ`p x n q converges toward Ȓ`p Dq. By compactness, we can assume that px n q nPN converges toward some x P C. Passing to the limit in ν C pexp xn p2 Ȓ`p x n qν C px n qqq " ´d dr exp xn prν C px n qq| r"2 Ȓ`p xnq , we obtain ν C pexp x p2 Ȓ`p Dqν C pxqqq " ´d dr exp x prν C pxqq| r"2 Ȓ`p Dq . In particular Ȓ`p Dq ą 0, otherwise we would end up with ν C pxq " ´νC pxq. As a consequence, we get Ȓ`p xq ď Ȓ`p Dq and finally Ȓ`p Dq " Ȓ`p xq, namely the infimum defining Ȓ`p Dq is attained and is positive. Then the mapping ψ C, Ȓ`p Dq is not injective, since

ψ C, Ȓ`p Dq pxq " exp x p Ȓ`p Dqν C pxqq " ψ C, Ȓ`p Dq pexp x p2 Ȓ`p Dqν C pxqqq
where x is still a minimizer in the definition of Ȓ`p Dq. Thus we get p R `pDq ď Ȓ`p Dq. Next, assuming that p R `pDq ă r R `pDq, let us show conversely that p R `pDq ě Ȓ`p Dq. Indeed, we can find distinct x, x 1 P C and r P p0, r R `pDqq such that ψ C,r pxq " ψ C,r px 1 q. Since r P p0, r R `pDqq, we can find a neighborhood A of x (respectively A 1 of x 1 , disjoint from A) in C such that ψ C,r is a diffeomorphism of A (resp. A 1 ) on its image. If the tangent space T ψ C,r pxq ψ C,r pAq of ψ C,r pAq at ψ C,r pxq is not equal to the tangent space T ψ C,r px 1 q ψ C,r pA 1 q of ψ C,r pA 1 q at ψ C,r px 1 q, then ψ C,r pAq and ψ C,r pA 1 q are crossing each other at ψ C,r pxq. Then by decreasing a little r into r 1 ă r, ψ C,r 1 pAq and ψ C,r 1 pA 1 q are still crossing each other. One can then find y P A and y 1 P A 1 such that ψ C,r 1 pyq " ψ C,r 1 py 1 q P ψ C,r pAq X ψ C,r 1 pA 1 q. This is in contradiction with the definition of p R `pDq. Thus we get T ψ C,r pxq ψ C,r pAq " T ψ C,r px 1 q ψ C,r pA 1 q. Note that by parallel transport along the geodesic, d dr exp x prν C pxqq is orthogonal to T ψ C,r pxq ψ C,r pAq and similarly for d dr exp x 1 prν C px 1 qq. It follows that the two unit vectors d dr exp x prν C pxqq and d dr exp x 1 prν C px 1 qq are proportional. They cannot be equal, otherwise by reversing time in the geodesics, we would end up with x " x 1 . So d dr exp x prν C pxqq " ´d dr exp x 1 prν C px 1 qq and by considering the geodesic starting from ψ C,r pxq with speed d dr exp x prν C pxqq and its reversed time geodesic, we get exp x p2rν C pxqq " x 1 and d ds exp x psν C pxqq| s"2r " ´νC px 1 q, namely r ě Ȓ`p Dq and as a consequence, p R `pDq ě Ȓ`p Dq, i.e. p R `pDq " Ȓ`p Dq.

We now come to the specific situation of the Euclidean plane.

Lemma 48 Assume that V " R 2 , endowed with its usual Euclidean structure. For any D P D and r P pR ´pDq, R `pDqq, we have

@ x P C, ρ ΨpC,rq pψ C,r pxqq " ρ C pxq 1 `rρ C pxq
In the context of Lemma 13, if α is given by @ x P C, αpxq " ρ C pxq 1 ´rρ C pxq then we have @ x P ΨpC, rq, T D Ψp¨, rqrαspxq " ρ ΨpC,rq pxq Proof One way to compute the curvature ρ ΨpC,rq pψ C,r pxqq, for x P C, is to consider a parametrization pypsqq s of ΨpC, rq by its length such that yp0q " ψ C,r pxq. The quantity ρ ΨpC,rq pψ C,r pxqq is then obtained by specializing the following formula at s " 0, B s τ ΨpC,rq pypsqq " ´ρΨpC,rq pypsqqν ΨpC,rq pypsqq where τ ΨpC,rq pypsqq is the unit vector B s ypsq.

Let pxpsqq s be a parametrization of C by its length, with xp0q " x. A parametrization of ΨpC, rq is then given by pψ C,r pxpsqqq s , but it is not by its length, due to the relation

B s ψ C,r pxpsqq " p1 `rρ C pxpsqqqτ C pxpsqq
To get a parametrization by the length, consider the time change pθ s q s given by ż θs 0 1 `rρ C pψ C,r pxpuqqq du " s and define ypsq " ψ C,r pxpθ s qq. We compute

B s ypsq " T ψ C,r rτ C pxpθ s qqsB s θ s " τ C pxpθ s qq
which is a unitary vector. We are thus led to differentiate

B s τ C pxpθ s qq " ´ρC pxpθ s qqν C pxpθ s qqB s θ s " ´ρC pxpθ s qq 1 `rρ C pψ C,r pxpsqqq ν C pxpθ s qq
This computation proves that ρ ΨpC,rq pypsqq " ρ C pxpθ s qq 1 `rρ C pψ C,r pxpsqqq (and that ν ΨpC,rq pypsqq " ν C pxpθ s qq, but that was already clear), which at s " 0 is the first assertion of the above lemma.

For the second one, note that for any D P D and r P pR ´pDq, R `pDqq, we have @ x P ΨpC, rq, ψ ´1 C,r pxq " ψ ΨpC,rq,´r pxq (note that r P pR ´pDq, R `pDqq implies that ´r P pR ´pΨpD, rqq, R `pΨpD, rqqq). It follows that for " ρ ΨpC,rq pxq

x P C, αpψ ´1 C,r pxqq " ρ C pψ ´1 C,r pxqq 1 ´rρ C pψ ´1 C,

Euclidean spaces

Let V " R n , fix x 0 P R n and r 0 ą 0 and consider the initial condition r G 0 " Bpx 0 , r 0 q and C 0 " B r G 0 . According to Lemma 48 (also by direct computation) we have for all r ą ´r0 , ρ ΨpC 0 ,rq pψ C 0 ,r pxqq " pn ´1q

1 r 0 1 `r r 0 " n ´1 r `r0 hpΨpD 0 , rqq " 2n r `r0 so @ x P C 0 , r α C 0 ,r pxq " n `1 r `r0
Since the above quantity does not depend on x, the solution of ( 95) is radial and r G t " Bpx, r R t q. According to (95), the radius starts with r R 0 " r 0 and its evolution is described by

@ t P r0, τq, d r R t " n `1 r R t `?2B t dt (97)
this equation being well-defined up to the stopping time

τ inftt ě 0 : r R t " ´?2B t or r R t " 0u
The condition r R t ą 0 comes from the fact that the normal flow ΨpC, rq is not defined when C is reduced to a singleton, and the condition r R t ą ´?2B t comes from the fact that the normal flow ΨpBBpx 0 , r R t q, rq is well-defined only for r ą ´r R t .

We get the following equation:

@ t P r0, τq, dp r R t `?2B t q " n `1 r R t `?2B t dt `?2dB t so p r R t `?2B t q tě0 " pBes pn`2q 2t
pr 0 qq tě0 , where Bes pn`2q pr 0 q pBes pn`2q t pr 0 qq tě0 is a Bessel process of dimension n `2 ě 2 starting from r 0 ą 0. For all t ě 0, r R t `?2B t ą 0, so pd r R t q{pdtq ą 0 and r R t ě r 0 ą 0, hence Equation (97) is well-defined for all times, i.e. τ " 8, and

@ t ě 0, D t " Ψp r G t , ? 2B t q " Bpx 0 , r R t `?2B t q
Since 0 is a entrance boundary for the Bessel process of dimension n `2, it is possible to solve the martingale problem associated to the generator pD, Lq and to the initial singleton condition D 0 " tx 0 u as follow: let Bes pn`2q p0q be a Bessel process of dimension n `2 starting at 0, and pB t q tě0 be the associated Brownian motion, namely such that @ t ě 0, Bes ˆ?2 ż t 0 ˆżBDs f dµ ˙dB s ˙tě0 is a martingale. We get for all t ě s ą 0,

F f pD t q ´Ff pD s q " ż t s LrF f spD u q du `Mt ´Ms (98)
Since a.s. lim sÑ0 `Ff pD s q " 0 and lim sÑ0 LrF f spD s q " " 0 , if n ě 3 8πf px 0 q , if n " 2 we can pass to the limit in (98) to get pD t q tě0 solves the martingale problem associated to the generator pD, Lq and to the singleton initial condition D 0 " tx 0 u.

Let us now consider the Doss-Sussman method relative to the decomposition [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF], for simplicity only in the illustrative Euclidean plane V " R 2 . For x 0 P R 2 and r 0 ą 0, we are interested in the initial condition D 0 Bpx 0 , r 0 q. Starting with pθ 0 , G 0 q " p0, D 0 q, we solve the evolution equation system (51) with respect to pθ t , G t q tPr0,τr 0 q . The solution pG t q tPr0,τr 0 q remains radial, so let us write it as G t " Bpx, Rt q for all t P r0, τ r 0 q. Equation (51) becomes:

@ t P r0, τ r 0 q, # d Rt " ´1 Rt`?2Bt`θt dt, R0 " r 0 dθ t " 4 Rt`?2Bt`θt dt, θ 0 " 0 (99)
where τ r 0 " inftt ě 0, Rt " 0 or ? 2B t `θt " ´R t u It follows that p Rt `?2B t `θt q tPr0,τr 0 q " pBes p4q 2t pr 0 qq tPr0,τr 0 q where Bes p4q pr 0 q is a Bessel process of dimension 4 starting from r 0 We deduce that τ r 0 " inftt ě 0, Rt " 0u and for any t P r0, τ r 0 q, Rt ´r0 " ´ż t 0 1 Rs `?2B s `θs ds

" ´1 2
ż 2t 0 1 Bes p4q
s pr 0 q ds Using the iterated logarithm law for the Bessel process for large times, we get

ż `8 0 1 Bes p4q s pr 0 q ds " ` 8
It follows that necessarily, a.s. τ r 0 ă 8 and more precisely that 2r 0 "

ż 2τr 0 0 1 Bes p4q s pr 0 q ds ( 100
)
Taking into account that for any t ą 0, we have (a.s.)

ż t 0 1 Bes p4q s p0q ds P p0, `8q
we can let r 0 go to 0 `in (100) to see that lim r 0 Ñ0

`τr 0 " 0 Thus, the Doss-Sussman method relative to the decomposition [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF] does not enable to define the dual process for all times nor permits approximations of singleton initial condition, contrary to the Doss-Sussman method associated to the decomposition (57).

Remark 52 It may be surprising at first view that several decompositions of a generator lead to solutions defined on different time domains. This is due to the fact that the flows associated to the corresponding vector fields may not be defined for all times. To get a simple example on R `, consider the case n " 1 in this subsection. 63

Hyperbolic spaces

Let V " H n be the hyperbolic space of dimension n. Fix some x 0 P H n and r 0 ą 0 and consider the initial condition r G 0 " D 0 Bpx 0 , r 0 q, and C 0 " B r G 0 . We have for any r ą ´r0 , ρ ΨpC 0 ,rq pψ C 0 ,r pxqq " pn ´1q cothpr `r0 q hpΨpD 0 , rqq " 2 sinh n´1 pr `r0 q Jpr `r0 q hence @ x P C 0 , r α C 0 ,r pxq " 2 sinh n´1 pr `r0 q Jpr `r0 q ´pn ´1q cothpr `r0 q where @ r ě 0, Jprq "

ż r 0 sinh n´1 puq du
The solution of (95) is radial, say r G t " Bpx, r R t q, and we have, starting with r R 0 " r 0 :

@ t P r0, τq, d r R t " ˜2 sinh n´1 p r R t `?2B t q Jp r R t `?2B t q ´pn ´1q cothp r R t `?2B t q ¸dt (101)
where,

τ " inftt ě 0 : r R t " ´?2B t or r R t " 0u
We get, for all t P r0, τq, dp r R t `?2B t q " ˜2 sinh n´1 p r R t `?2B t q Jp r R t `?2B t q ´pn ´1q cothp r R t `?2B t q ¸dt `?2dB t Note that as r ą 0 goes to zero,

2 sinh n´1 prq Jprq ´pn ´1q cothprq " n `1 r
This behavior is sufficient to insure that 0 is an entrance boundary for the diffusion p r R t `?2B t q tě0 (see for instance the classical computations of Chapter 15 of Karlin and Taylor [START_REF] Karlin | A second course in stochastic processes[END_REF]). In particular, since p r R t `?2B t q tě0 starts from r 0 ą 0, it will never reach 0 (a.s.). Furthermore, let us check that the radius process p r R t q tě0 of p r G t q tě0 is non-decreasing. Indeed, after an integration by parts, we obtain for all r ě 0:

ż r 0 sinh n´1 puq du " ż sinhprq 0 v n´1 ? 1 `v2 dv " sinh n prq n coshprq `ż sinhprq 0 v n`1 n ? 1 `v2 p1 `v2 q dv ď sinh n prq n coshprq `1 n ż sinhprq 0 v n´1 ? 1 `v2 dv.
Hence we have for any r ě 0, This non-negativity and (101) show that p r R t q tě0 is non-decreasing. From these observations, we get the solution of ( 101) is defined for all times, i.e. τ " 8, and finally

@ t ě 0, D t " Bpx 0 , r R t `?2B t q
provides a solution to the martingale problem associated to the generator pD, Lq and starting from Bpx 0 , r 0 q. As in the Euclidean case, by letting r 0 go to zero, we solve the martingale problem associated to the generator pD, Lq starting from the singleton tx 0 u.

Spherical spaces

Let V " S n be the sphere of dimension n P N. Fix x 0 P S n and r 0 P p0, πq, and consider the initial condition r G 0 " Bpx 0 , r 0 q, and C 0 " B r G 0 . We have for any r P p´r 0 , π ´r0 q (note that the normal flow in not well-defined for all positive times): r α C 0 ,r pxq " 2 sin n´1 pr `r0 q Ipr `r0 q ´pn ´1q cotpr `r0 q

where Ipsq " ş s 0 sin n´1 puq du, for any s P r0, πs. The solution of (95) is radial, say r G t " Bpx, r R t q. According to (95), starting from r R 0 " r 0 , we have

@ t P r0, τq, d r R t " ˜2 sin n´1 p r R t `?2B t q Jp r R t `?2B t q ´pn ´1q cotp r R t `?2B t q ¸dt ( 102
)
where

τ " inftt ě 0 : r R t " π ´?2B t or r R t " ´?2B t or r R t " 0u
We get @ t P r0, τq, dp r R t `?2B t q " ˜2 sin n´1 p r R t `?2B t q Ip r R t `?2B t q ´pn ´1q cotp r R t `?2B t q ¸dt `?2dB t Again, we have as r goes to 0 `, 2 sin n´1 prq Iprq ´pn ´1q cotprq " n `1 r and this behavior is sufficient to get that 0 is an entrance boundary for the diffusion p r R t `?2B t q tě0 . It follows that it never hits 0. To show that p r R t q tě0 is non-decreasing, let us check that @ r P p0, πq, 2 sin n´1 prq Iprq ´pn ´1q cotprq ě 0

Observe that it is clearly satisfied for r P r π 2 , πq. For r P p0, π 2 q, we have: In fact τ is the hitting time of the whole sphere S n by pD t q tPr0,τs . Since for all f P C 8 pS n q, we have LrF f spS n q " 0, it is natural to let the latter process be absorbed at S n , namely to extend it by @ t ě τ, D t S n so that pD t q tě0 provides a solution to the martingale problem associated to the generator pD, Lq and starting from Bpx 0 , r 0 q. As in the Euclidean and hyperbolic cases, the martingale problem associated to the generator pD, Lq and starting from the singleton tx 0 u is solved by letting r 0 go to zero.

ż r 0 sin n´1 puq du " ż sinprq 0 v n´1 ? 1 ´v2
6 About the martingale problems associated to L After proving Theorem 5, we will show that the martingales naturally associated to L are directed by a unique Brownian motion, property corresponding to the radial evolution [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. Next, we will enrich the set of elementary observables and see in the particular example of the diffusion X consisting of the Brownian motion in the Euclidean plane how the enriched martingale problem is sufficient to deduce that the dual domain-valued process ends up looking like a big disk, at least if it can be defined for all times.

Proof of Theorem 5

As explained above Theorem 5, we assume we are given a stochastic process pD t q tPr0,τq taking values in G for positive times and solution to the martingale problem associated to pD, Lq, defined as in the introduction, except that the elementary observables are defined on G instead of D. Despite this generalization, the following arguments are similar to those given in the one-dimensional case treated in [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF].

Let a test function f P C 8 pR `q be given and consider the process pS t q tPr0,τq defined by @ t P r0, τq, S t fpµpD t qq " fpF 1 pD t qq

Since the mapping G Q D Þ Ñ fpF 1 pDqq belongs to D, there exists a local martingale pM t q tPr0,τq such that for all t P r0, τq,

S t " S 0 `ż t 0 Lrf ˝F1 spD s q ds `Mt (103)
By definition of L, we have

Lrf ˝F1 spDq " f 1 pF 1 qLrF 1 s `f2 pF 1 qΓ L rF 1 , F 1 s
Recall that in the proof of Theorem 3, we computed, for any D P G, with C BD,

LrF 1 spDq " 2 µpCq 2 µpDq (104) Γ L rF 1 , F 1 spDq " µpCq 2 (105) so that Lrf ˝F1 spDq " µpCq 2 ˆf2 pF 1 q `2 f 1 pF 1 q F 1 ˙pDq " 2µpCq 2 LrfspF 1 pDqq
where

@ x P R ˚, L 1 2 B 2 `1 x B
is the generator of the Bessel process of dimension 3 on R `(see e.g. Chapter 11 of the book [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF] of Revuz and Yor). Thus we obtain, for all t P r0, τq, S t " S 0 `2 ż t 0 µpC s q 2 LrfspµpD s qq ds `Mt It leads us to introduce the time change described by ( 12) and ( 13) and @ t P r0, ςq, R t µpD θptq q to get pR t q tPr0,ςq is a stopped continuous solution to the martingale problem associated to the generator pC 8 pR `q, Lq. It follows that pR t q tPr0,ςq is a stopped Bessel process of dimension 3. For completeness, let us just recall the underlying argument.

Define for t P r0, ςq,

W t R t ´R0 ´ż t 0 1 R s ds
According to the martingale problem, the process pW t q tPr0,ςq is a continuous local martingale whose bracket is given by @ t P r0, ςq, xW y t "

ż t 0 Γ L rid, idspR s q ds
where Γ L is the carré du champ operator associated to L and id : R ˚Q x Þ Ñ x is the identity mapping on R ˚. Since Γ L rid, ids " pid 1 q 2 " 1, we get @ t P r0, ςq, xW y t " t so Lévy's theorem shows that pW t q tPr0,ςq is a stopped Brownian motion. Then pR t q tPr0,ςq is solution to the stochastic differential solution @ t P r0, ςq, dR t " dW t `1 R t dt which admits a unique strong solution, once R 0 is given. In particular the law of pR t q tPr0,ςq is determined by the initial distribution of R 0 , it is the Bessel process of dimension 3 with initial law LpX 0 q.

The stochastic differential equation associated with the martingale problem

With the notation of the above proof, for f " id in (103), we get M θt " W t for t P r0, εq, or

@ t P r0, τq, M t " W θ ´1 t
where θ ´1 : r0, τq Ñ r0, εq is the inverse mapping of θ given in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]. In particular, we get

@ t P r0, τq, xM y t " θ ´1 t " 2
ż t 0 µpC s q 2 ds so that we can find a Brownian motion pB t q tě0 (up to enlarging the underlying probability space) such that @ t P r0, τq, M t " ? 2 ż t 0 µpC s q dB s Namely we have @ t P r0, τq, dµpD t q " 2 µpC t q 2 µpD t q dt `?2µpC t q dB t (106)

The same Brownian motion pB t q tě0 is driving all the pF f pD t qq tPr0,τq , for all f P C 8 pV q, and even more:

Proposition 53 For all F P D, we have @ t P r0, τq, FpD t q " FpD 0 q `ż t 0 LrFspD s q ds `?2

ż t 0 a Γ L rF, FspD s q dB s (107)
where the determination of the sign of a Γ L rF, Fs is

@ D P D, a Γ L rF, FspDq ÿ lP n B l fpF f 1 , ..., F fn qpDq ż C f l dσ
when F " fpF f 1 , ..., F fn q, with the notation of the introduction.

Proof

By definition of pD, Lq and due to the usual rules of continuous stochastic calculus (see for instance the book [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF] of Revuz and Yor), it is sufficient to check the above formula on the elementary observables, namely that for all f P C 8 pV q, @ t P r0, τq, F f pD t q " F f pD 0 q `ż t 0 LrF f spD s q ds `?2

ż t 0 b Γ L rF f , F f spD s q dB s
with the determination of sign:

a Γ L rF f , F f s ş f dσ.
From the martingale problem, we know that for any f P C 8 pV q, the process @ t P r0, τq, M f t F f pD t q ´Ff pD 0 q ´ż t 0 LrF f spD s q ds is a local martingale whose bracket is given by @ t P r0, τq,

@ M f D t 2 ż t 0 Γ L rF f , F f spD s q ds
So our goal is to check that @ t P r0, τq,

M f t " ż t 0 a Γ L rF f , F f s a Γ L rF 1 , F 1 s pD s q dM 1 s
Since all the considered martingales start from 0, it is equivalent to show that @ t P r0, τq,

C M f ¨´ż 0 a Γ L rF f , F f s a Γ L rF 1 , F 1 s pD s q dM 1 s G t " 0
Developing by polarization the l.h.s., we obtain

@ M f D t `Cż 0 a Γ L rF f , F f s a Γ L rF 1 , F 1 s pD s q dM 1 s G t ´2 C M f , ż 0 a Γ L rF f , F f s a Γ L rF 1 , F 1 s pD s q dM 1 s G t " @ M f D t `ż t 0 Γ L rF f , F f s Γ L rF 1 , F 1 s pD s q d @ M 1 D s ´2 ż t 0 a Γ L rF f , F f s a Γ L rF 1 , F 1 s pD s q d @ M f , M 1 D s " 2 ż t 0 Γ L rF f , F f spD s q ds `2 ż t 0 Γ L rF f , F f s Γ L rF 1 , F 1 s pD s q Γ L rF 1 , F 1 spD s qds ´4 ż t 0 a Γ L rF f , F f s a Γ L rF 1 , F 1 s pD s q Γ L rF f , F 1 spD s qds " 4 ż t 0 ˜ΓL rF f , F f s ´aΓ L rF f , F f s a Γ L rF 1 , F 1 s Γ L rF 1 , F f s ¸pD s q ds " 0
where we used that for any D P G,

˜ΓL rF f , F f s ´aΓ L rF f , F f s a Γ L rF 1 , F 1 s Γ L rF 1 , F 1 s ¸pDq " ˆżC f dµ ˙2 ´şC f dµ µpCq µpCq ż C f dµ " 0
Remark 54 The stopped standard Brownian motion pB t q tPr0,τq in ( 107) is (a.s.) on the random interval r0, τq, the same as the one appearing in Theorem 17, when above, one considers the stochastic process pD t q tPr0,τq constructed in Theorem 17. This is a consequence, on one hand of (106), which enables to recover pB t q tPr0,τq from pD t q tPr0,τq , since B 0 " 0 and µpC t q ą 0 for t P r0, τq, and on the other hand of the fact that in the proof of Theorem 17, we have @ t P r0, τq, M t " ? 2 ż t 0 ˆżCs f dµ ˙dB s so by taking f " 1, we can recover pB t q tPr0,τq in the same way. In the same spirit as Theorem 5 and similarly to [START_REF] Miclo | Strong stationary times for one-dimensional diffusions[END_REF], we also have Proposition 55 Under the setting of Theorem 5, the process p1{µpD t qq tPr0,τq is a positive local martingale. It follows that lim tÑτ´µ pD t q exists a.s. in p0, `8s.

Proof

Consider the mapping F : G Q D Þ Ñ 1{µpDq, which belongs to D. To see that p1{µpD t qq tPr0,τq is a local martingale, it is sufficient to check that LrFs " 0. By definition,

@ D P G, LrFspDq " ´1 F 2 1 pDq LrF 1 spDq `2 F 3 1 pDq Γ L rF 1 , F 1 spDq " ´1 F 2 1 pDq µpCq 2 µpDq `2 F 3 1 pDq µpCq 2
" 0 where (104) and (105) were taken into account. Thus as a positive submartingale 1{µpD t q, converges a.s. as t goes to τ from below, to a limit belonging to r0, `8q. By taking the inverse, we get the announced result.

Enrichment of the elementary observables

Up to now, we only considered elementary observables of type (4), since they were sufficient for our purposes, but other functionals are interesting to go further. To simplify the presentation, we restrict ourselves to the situation of the Brownian motion on a Riemannian manifold, namely we take b " 0, so that µ " λ, µ " σ and ρ b " ρ. The general case can be treated similarly (see the manipulations of the proof of Theorem 3).

The first of new elementary observables we would like to add have the following form, for any f P C 8 pV q,

G f : D Q D Þ Ñ G f pDq ż C f dσ (108)
Indeed, the action (6) of the generator L can then be rewritten, taking into account Stokes' theorem [START_REF] Miclo | On the construction of set-valued dual processes[END_REF], as

@ D P D, LrF f spDq " ż C x∇f, νy `2 σpCq σpDq f dσ " F f pDq `2G 1 pDqG f pDq F 1 pDq
so it seems natural to study the evolution of pG f pD t qq tPr0,τq , when pD t q tPr0,τq is a solution to the martingale problem associated to L.

Unfortunately, it seems difficult to work directly from this martingale problem, while we still don't know if it is well-posed. Our hope is that by enriching the domain of functionals to which it is applied, we should be more able to obtain that it is well-posed. So we rather consider the process pD t q tPr0,τq given by (52) and construct new martingales for it. More precisely, up to reducing τ (replacing it by its minimum with the first time D t is no longer included into a nice tubular neighborhood of D 0 ), we will assume that D τ is defined and belong to D. Before investigating the functionals of the form (108), we are interested in the composition of the process pD t q tPr0,τs with the normal flow, which already played a crucial role in the construction of pD t q tPr0,τs . So define R tr P R : @ t P r0, τs, D t P D r u @ r P R, @ t P r0, τq,

D prq t ΨpD t , rq (109)
" ΨpG t , ? 2B t `θt `rq where pG t q tPr0,τs and pθ t q tPr0,τs are defined as in (51). For any r P R, consider @ D P D r , @ x P C, α prq C pxq ρ C pxq ´ρΨpC,rq pψ C,r pxqq and the operator L prq acting on D ´r via @ f P C 8 pV q, @ D P D ´r, L prq rF f spDq "

ż C x∇f, νy `ˆ2 σpΨpC, ´rqq λpΨpD, ´rqq `αp´rq C ˙f dσ " ż D f dλ `2σpΨpC, ´rqq ş C f dσ λpΨpD, ´rqq `żC α p´rq C f dσ (110)
Its interest comes from:

Lemma 56 For any f P C 8 pV q, t P r0, τs and r P R, we have

F f pD prq t q " F f pD prq 0 q `ż t 0 L prq rF f spD prq s q ds `?2 ż t 0 G f pD prq s q dB s

Proof

The arguments are similar to those of the proof of Theorem

" ´Br α prq C pxq| r"0 " B r α p´rq C pxq| r"0 By differentiation with respect to r at 0 in Lemma 56, we get: Proposition 57 For any f P C 8 pV q, we have @ t P r0, τs, G f pD t q " G f pD 0 q `ż t 0 LrG f spD s q ds `?2 Note that for any f P C 8 pV q and D P D, we have

ż
LrG f spDq " G f pDq `2 G 1 pDq F 1 pDq F f pDq `2 G 1 pDq F 1 pDq ż C ρf dσ `żC ρ p1q f dσ but neither ş C ρf dσ nor ş C ρ p1q
f dσ are of the form F g of G g for some g P C 8 pV q. We are thus lead to introduce two new types of elementary observables:

H f : D Q D Þ Ñ H f pDq ż C ρf dσ H p1q f : D Q D Þ Ñ H f pDq ż C ρ p1q f dσ
Investigating the evolution of these observables, one will have to consider more generally for any l P Z

H plq f : D Q D Þ Ñ H f pDq ż C ρ plq f dσ (112)
where by iteration, for any n P Z `, @ x P C, ρ Probably other functionals will also appear (such as

D Q D Þ Ñ ş C ρ xν, ∇f y dσ or D Q D Þ Ñ ş C ρ 2 f
dσ, see the next lemma), but the study of these iterations, as well as their impact on the well-posedness of the corresponding martingale problems, is left for a future work.

In the same spirit, we remark that the introduction of ρ p1q and H p1q are already needed to consider a third derivative in Lemma 10:

Lemma 58 For any f P C 8 pV q and D P D, we have and by differentiating with respect to r at 0.

The case f " 1 is particularly interesting, since G 1 pDq " σpCq for any D P D. The quantity ş C ρ dσ is called the total mean curvature of C and according to the previous lemma, ş C ρ p1q `ρ2 dσ is the derivative of the total mean curvature along the normal radial flow. In the situation of constant curvature in dimension 2, the terms ρ p1q and ρ 2 are in fact comparable: Lemma 59 Assume that V is a surface of constant curvature K P R. Then we have @ D P D, ρ p1q " ´ρ2 ´K Proof When V is the Euclidean plane, the result follows by differentiating at r " 0 the first formula given in Lemma 48. The other null curvature situations (cylinders and flat torus) can be treated similarly, since they can be up-lifted to their locally isometric covering R 2 .

For the other constant curvature cases, use instead Lemma 50 of Subsection 5.1.

Remark 60 (a) When V is the Euclidean plane, it follows from Lemma 59 that B r ż ΨpC,rq ρ dσ " 0 namely locally the normal radial flow leaves the total curvature of a smooth curve invariant. This is in fact a consequence of Hopf's Umlaufsatz Theorem, stating that for any piecewise differentiable curve

C in R 2 ż C ρ dσ " 2π (113)
(with an appropriate convention for the jumps of the tangent vectors, where ρ dσ has to be seen as the difference of angles times a Dirac mass at the considered singular point The last two formulas are valid on any Riemannian manifold V of dimension n.

But when V has a constant sectional curvature K, since

@ x P C, @ m P n ´1 , λ p1q m,C pxq " ´K ´λ2 m,C
we obtain that, at least if none of the eigenvalues vanishes,

@ x P C, κ p1q C pxq " ´¨ρ C pxq `K ÿ mP n´1 1 λ m,C pxq 'κ C pxq
Integrating this relation with respect to σ on C, it follows from (114) that

B r ż ΨpC,rq κ C dσ ˇˇˇr "0 " ´K ż C ÿ mP n´1 1 λ m,C κ C dσ When n " 3, we have ˆ1 λ 1,C `1 λ 2,C ˙κC " λ 1,C `λ2,C " ρ C thus B r ż ΨpC,rq κ C dσ ˇˇˇr "0 " ´K ż C ρ C dσ " ´K B r λpΨpD, rqq| r"0
Namely the quantity

ż C κ C dσ `KλpDq
is invariant under the normal radial flow (as long as it remains in D). This is a very special case of the Gauss-Bonnet theorem, asserting that the above quantity is equal to 2π times the Euler characteristic of V . Again, one is left wondering about possible links between the normal radial flow and the generalized Gauss-Bonnet theorem.

(c) It is also natural to ask for a generalization of Lemma 59 when V is a surface whose curvature is not constant.

Let us come back to our martingale problem and to Proposition 57. The explicit description of the martingale associated to the evolution of pG f pD t qq tPr0,τs in terms of the stopped Brownian motion pB t q tPr0,τs , enables us to see that for any f, g P C 8 pV q and D P D, Γ L rF f , G g spDq " G f pDq pF g pDq `Hg pDqq Γ L rG f , G g spDq " pF f pDq `Hf pDqq pF g pDq `Hg pDqq These formulas leads to an enrichment of the algebra D of the introduction. Indeed, consider the new algebra D consisting of the functionals of the form F fpA 1 , ..., A n q, where n P Z `, A 1 , ..., A n are elementary observables of the form (4) or (108) and f : R Ñ R is a C 8 mapping, with R an open subset of R n containing the image of D by pA 1 , ..., A n q. For such a functional F, define LrFs "

ÿ jP 1,n B j fpA 1 , ..., A n qLrA j s `ÿ k,lP 1,n B k,l fpA 1 , ..., A n qΓ L rA k , A l s
To two elements of D, F fpA 1 , ..., A n q and G gp r A 1 , ..., r A m q, we also associate

Γ L rF, Gs ÿ lP n ,kP m B l fpA 1 , ..., A n qB k gp r A 1 , ..., r A m qΓ L rA l , r A k s
These formulas can be directly obtained as consequences of Itô's formula applied to the expressions given in (107) and Proposition 57, since the corresponding Brownian motions are the same (cf. Remark 54).

Asymptotic behavior for large times on the plane

In this last subsection, we present an example of application of the above extension of the domain of L. We consider the Laplacian L " on the Euclidean plane R 2 . We assume the domain of L has been extended to contain all mappings of the forms ( 4) and ( 108), defined on G, an extension of D as described before Theorem 5. Just make the hypothesis that the boundaries of the elements of G are piecewise differentiable curves.

Theorem 61 Let pD t q tě0 be a solution to the martingale problem associated to L defined for all times. Then we have a.s. in the Hausdorff metric, lim tÑ`8 D t a λpD t q " Bp0, 1{ ? πq where Bp0, 1{ ? πq is the Euclidean ball centered at 0 of radius 1{ ? π.

Proof

From Theorem 5, we know that for any t ą 0, λpD t q ą 0, namely D t is not a singleton and belongs to G by assumption. Up to replacing pD t q tě0 by pD 1`t q tě0 , we assume in this proof that D t belongs to G for all t ě 0.

In the Euclidean plane, the following isoperimetric inequality holds:

@ D P G, σpCq 2 λpDq ě 4π (115)
with equality if and only if D is a ball. From Proposition 55 and τ " `8, we deduce that lim inf tÑ`8 σpC t q ě 2 lim tÑ`8 a πλpD t q ą 0 Thus in [START_REF] Huang | The Cauchy problem for fully nonlinear parabolic systems on manifolds[END_REF] we get ς " `8 and in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], lim tÑ`8 θ t " `8.

In these circunstances, Theorem 5 asserts that pλpD θt qq tě0 is a Bessel process of dimension 3 and in particular lim tÑ`8 λpD t q " `8

We now use Proposition 57. From the relation G 1 pDq " σpCq, we get in general that dσpC t q " ˆżCt ρ p1q `2 σpC t q λpD t q ρ dσ ˙dt `?2 ˆżCt ρ dσ ˙dB t But for the Euclidean space, we have ρ p1q " ´ρ2 and ş ρ dσ " 2π, according to Lemma 59 and Hopf's Umlaufsatz Theorem (113) (taking into account that the considered boundaries are piecewise differentiable), respectively. Thus we get dσpC t q " ˆ´ż Ct ρ 2 dσ `4π σpC t q λpD t q ˙dt `2? 2π dB t and dσpC t q 2 " 2 ˆ´ż Ct ρ 2 dσ `4π σpC t q λpD t q ˙σpC t q dt `4? 2πσpC t q dB t `8π 2 dt

Recall from (106) that dλpD t q " 2 σpC t q 2 λpD t q dt `?2σpC t q dB t Consider the process Z pZ t q tě0 defined by @ t ě 0, Z t σpC t q 2 ´4πλpD t q

From the above computations, we deduce that @ t ě 0, dZ t " 2 ˆ4π 2 ´σpC t q ż Ct ρ 2 dσ ˙dt By Cauchy-Schwarz' inequality, we have for any t ě 0,

4π 2 " ˆżCt ρ dσ ˙2 ď σpC t q ż Ct ρ 2 dσ
showing that Z is a.s. non-increasing. Thus we have

@ t ě 0, Z t ď Z 0 (116)
For any t ě 0, denote r D t D t { a λpD t q. We have for any t ě 0, σp r C t q 2 ´4πλp r D t q " σpC t q 2 ´4πλpD t q λpD t q ď σpC 0 q 2 ´4πλpD 0 q λpD t q and the last expression goes to zero as t goes to `8. From Bonnesen's inequality (see e.g. the book of Burago and Zalgaller [START_REF] Yu | Geometric inequalities, volume 285 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]), we deduce that as t goes to infinity, r D t becomes closer and closer, in Hausdorff metric, to a disk of volume 1. To see the announced result, it is sufficient to see that the barycenter of r D t , which is the barycenter of D t divided by a λpD t q, i.e.

1 λpD t q 3{2 ż Dt
x λpdxq converges a.s. to 0 as t goes to `8. It amounts to see that F f {F 3{2 1 pD t q converges to zero for t large, where f is either the first or the second canonical projection of R 2 . So let f be the first coordinate mapping (the second coordinate can be treated similarly, note that a symmetry argument cannot be used here, since the well-posedness is missing). Before investigating the evolution of R `Q t Þ Ñ F f {F 3{2 1 pD t q, we need a preliminary result.

Lemma 62 A transition phenomenon occurs:

@ a ą 1, ż `8 0 1 λpD t q a ds ă `8 while @ a ď 1, ż `8 0 1 λpD t q a ds " ` 8
Furthermore, we have for large t ě 0, a.s.,

ż t 0 1 λpD s q ds " lnpF 1 qpD t q 4π
Proof This is based on the fact that λpD t q goes to infinity as t goes to infinity. More precisely, taking into account ( 104) and (105), we compute, for any a ą 0 and any D P G,

L " 1 F a 1  pDq " ´a F a`1 1 pDq LrF 1 spDq `apa `1q F a`2 1 Γ L rF 1 , F 1 spDq " apa ´1q σpCq 2 λpDq a`2
and in the sense of Proposition 53

a Γ L " 1 F a 1  pDq " ´a F a`1 1 pDq G 1 pDq " ´aσpCq λpDq a`1 where ? Γ L r1{F a 1 s stands for a Γ L r1{F a 1 , 1{F a 1 s.
Since for any a ą 0, we know that 1{F a 1 pD t q converges to zero as t goes to infinity, we deduce that

1 F a 1 pD t q ´1 F a 1 pD 0 q " ż t 0 L " 1 F a 1  pD s q ds `?2 ż t 0 a Γ L " 1 F a 1  pD s q dB s " apa ´1q ż t 0 σpC s q 2
λpD s q a`2 ds ´?2a ż t 0 σpC s q λpD s q a`1 dB s (117) converges for large t ě 0. By a contradictory argument, assume that ż `8 0 σpC s q 2 λpD s q 2a`2 ds " `8

which implies in particular that ż `8 0 σpC s q 2 λpD s q a`2 ds " `8

since lim tÑ`8 λpD t q " `8. The bracket of the local martingale p ş t 0 ? Γ L r 1 F a 1 spD s q dB s q tě0 is given for any t ě 0 by

Bż 0 a Γ L " 1 F a 1  pD s q dB s F t " ż t 0 Γ L " 1 F a 1  pD s q ds " a 2
ż t 0 σpC s q 2 λpD s q 2a`2 ds so that the iterated logarithm law for continuous local martingales implies lim sup tÑ`8

ż t 0 a Γ L " 1 F a 1  pD s q dB s " `8 lim inf tÑ`8 ż t 0 a Γ L " 1 F a 1  pD s q dB s " ´8
In view of (118), it would follow that for large t ě 0, the expression in (117) admits ´8 as liminf if a ď 1 and `8 as limsup if a ě 1, this is in contradiction with the existence of a finite limit. Thus we get ż `8 0 σpC s q 2 λpD s q 2a`2 ds ă `8

We get the first announced result, remembering that for large t ě 0, σpC t q " 2 a πλpD t q. For the second result with a " 1, rather consider the observable lnp1{F 1 q. We have for any D P G,

L rlnpF 1 qs pDq " 1 F 1 pDq LrF 1 spDq ´1 F 2 1 pDq Γ L rF 1 spDq " σpCq 2 λpDq 2 and a Γ L rlnpF 1 qs pDq " 1 F 1 pDq G 1 pDq " σpCq λpDq
So via similar contradictory arguments as before with lnpF 1 qpD t q ´lnpF 1 qpD 0 q " ż t 0 σpC s q 2 λpD s q 2 ds ´?2

ż t 0 σpC s q λpD s q dB s (119)
which diverges to `8 as t goes to infinity, we end up with ż `8 0 σpC s q 2 λpD s q 2 ds " `8

For the last result, we need to apply more carefully the iterated logarithm law. Let pM t q tě0 be the continuous local martingale defined by

@ t ě 0, M t ż t 0 σpC s q λpD s q dB s
Its bracket is given by @ t ě 0, xM y t ż t 0 σpC s q 2 λpD s q 2 ds Since xM y t diverges to `8 for large t ě 0, the iterated logarithm law asserts that lim sup tÑ`8

|M t | a xM y t lnplnpxM y t qq " 1
It follows that for large t ě 0,

|M t | ! ż `8 0 σpC s q 2 λpD s q 2 ds
and the last statement of the lemma is a direct consequence of (119) and of the fact that σpC t q 2 " 4πλpD t q, for large t ě 0.

Let us come back to our objective to show that ξ t converges a.s. toward 0, where

@ t ě 0, ξ t F f pD t q F 3{2 1 pD t q with f the first coordinate mapping of R 2 . Instead of applying the martingale problem directly to the composed observable D Q D Þ Ñ F f {F 3{2 1 pDq, it seems more convenient to decompose ξ t into M t { a
λpD t q, where pM t q tě0 is defined by

@ t ě 0, M t F f F 1 pD t q " Λrf spD t q
From Theorem (3), we have

LrΛrf sspDq " Λr rf ss " 0 so it follows that pM t q tě0 is a local martingale. More precisely, we get from Proposition 53 that

@ t ě 0, M t " M 0 `ż t 0 h s dB s
where for any s ě 0,

h s a Γ L rF f {F 1 spD s q " G f F 1 pD s q ´Ff F 2 1 pD s qG 1 pD s q " G 1 F 1 pD s q ˆGf G 1 ´Ff F 1 ˙pD s q
When f is replaced by the identity mapping id : R 2 Ñ R 2 , for any D P G, the vector ´Gid

G 1
´Fid F 1 ¯pDq is the difference between the barycenter of C and the barycenter of D, so it appears easily that for any s ě 0,

|h s | ď σpC s q λpD s q › › › › G id G 1 pD s q ´Fid F 1 pD s q › › › › ď σpC s q 2 2λpD s q
More precise computations, separately presented in [START_REF] Miclo | Isoperimetric stability of boundary barycenters in the plane[END_REF] because they rely on techniques belonging to the field of isoperimetric stability, show that there exists a universal constant c ą 0 such that for any D P G with σpCq 2 ´4πλpDq ď λpDq{π, we have

› › › › G id G 1 pDq ´Fid F 1 pDq › › › › ď cλ 1{4 pDqpσpCq 2 ´4πλpDqq 1{4
Thus taking into account the decreasing property (116) and the fact that λpD s q diverges to `8 as s goes to infinity, we get there exists (a.s.) a random time S and a constant χ (depending on D 0 ) such that

@ s ě S, |h s | ď χ λpD s q 1{4
From the iterated logarithm law, we deduce that as t goes to `8,

|M t | " r O ˜dż t 0 1 a λpD s q ds ¸(120)
where the notation φptq " r Opϕptqq, for two functions φ, ϕ : R `Ñ R `with lim tÑ`8 ϕptq " `8, means that lim sup tÑ`8 φptq ϕptq lnplnpϕptqqq ă `8

Applying the martingale problem to the composed functional ? F 1 , we get that for any t ě 0, a F 1 pD t q " a F 1 pD 0 q `3 4 ż t 0 σpC s q 2 λpD s q 3{2 ds `1 ? 2 ż t 0 σpC s q a λpD s q dB s Using again, on one hand that σpC s q 2 and λpD s q are of the same order for large s ě 0, and on the other hand the iterated logarithm law, we deduce that for large t ě 0,

ż t 0 1 a λpD s q ds " r Op a λpD t q `?tq (121)
Another application of the iterated logarithm law to three independent Brownian motions enables to see that if pR t q tě0 is a Bessel process of dimension 3, then a.s.,

R t " r Op ? tq (122)
Recall that pR t q tě0 λpD θt q tě0 is a Bessel process of dimension 3, according to Theorem 5, where pθ t q tě0 is defined by @ t ě 0, 2 ż θt 0 σpC s q 2 ds " t

The martingale problem applied to F 1 shows that for any t ě 0, λpD t q " λpD 0 q `2 ż t 0 σpC s q 2 λpD s q ds `?2 ż t 0 σpC s q dB s Replacing t by θ t , we deduce that θ t " 1 4π ż θt 0 σpC s q 2 λpD s q ds " 1 8π ˆλpD θt q ´λpD 0 q ´?2 according to Theorem 3.2 (ii) of Shiga and Watanabe [START_REF] Shiga | Bessel diffusions as a one-parameter family of diffusion processes[END_REF], see also Motoo [START_REF] Motoo | Proof of the law of iterated logarithm through diffusion equation[END_REF] (the part (i) of their theorem extends (122) to any Bessel process with a positive parameter). This implies that lim tÑ`8 lnpR t q lnptq " 1 2

ż
Furthermore, note that in the above proof we did not use the last part Lemma 62, which also gives an equivalent of lnpλpD t qq for large t ě 0.

These shortcomings are an invitation to study further the asymptotic behavior of the renormalized domains pD t { a λpD t qq tě0 , in particular their fluctuations around the convergence of Theorem 61.

Elliptic density theorem revisited

Here we assume that Conjecture 6 is true: not only we can construct a solution pD t q tPr0,τs to the martingale problem associated to pD, Lq and starting from any singleton tx 0 u Ă V , but it can be coupled with the primal diffusion X starting from x 0 so that ( 14) and ( 15) are satisfied. Let us show how to quickly recover the density theorem for elliptic diffusion from this property.

The proof is based on the following elementary observation:

Lemma 64 Let A Ă V be a negligible event with respect to µ and denote f its indicator function. For any measurable D Ă V with µpDq ą 0 and s ě 0, we have ΛrP s rf sspDq " 0 where pP t q tě0 is the Markov semi-group associated to L, seen as a family of Markov kernels.

Proof

Taking into account that µ is invariant for pP t q tě0 , we have ΛrP s rf sspDq " µr1 D P s rf ss µpDq ď µrP s rf ss µpDq " µrf s µpDq " 0

We can now come to the

Proof of Corollary 7

With the notations of the above lemma and Corollary 7, we want to check that for any x 0 P V and any r ą 0, P r rf spx 0 q E x 0 rf pX r qs " 0.

For any t ě 0, let F t be the σ-field generated by X r0,ts and D r0,t^τs . From ( 14), we get the diffusion X pX t q tě0 is also strongly Markovian with respect to the filtration pF t q tě0 . Remark that τ ą 0, the stopping time entering into the definition of pD t q tPr0,τs , is also a stopping time with respect to pF t q tě0 . It follows that E x 0 rf pX r qs " E x 0 rE x 0 rf pX r q|F r^τ ss " E x 0 rP r´r^τ rf spX r^τ qs For any t ě 0, let D t be the σ-field generated by D r0,t^τs . It follows from [START_REF] Karlin | A second course in stochastic processes[END_REF] with T " r ^τ, that Erhpr ^τ, X r^τ q|D r^τ s " ΛpD r^τ , hpr ^τ, ¨qq for any non-negative measurable mapping h : R `ˆV Ñ R `. We deduce that E x 0 rf pX r qs " E x 0 rE x 0 rP r´r^τ rf spX r^τ q|D r^τ ss " E x 0 rΛrP r´r^τ rf sspD r^τ qs " 0 according to Lemma 64. Indeed, we took into account Theorem 5, insuring that for any t P p0, τs, we have µpD t q ą 0.

With Marc Arnaudon, we are currently working on the existence of a coupling as in Conjecture 6 and some results in this direction will be presented in a future paper.

When the solutions to the martingale problems associated to pD, Lq and to initial singleton sets can be defined for all times, there is no need to have such a coupling at our disposal to recover the density theorem for elliptic diffusions. Indeed, assume that for any x 0 P V , we can construct a solution pD t q tě0 to the martingale problem associated to pD, Lq and starting from the singleton tx 0 u Ă V . First, we remark that we can enrich the martingale problem by adding a temporal component. Let us just sketch the argument: when F P D and f P C 1 pr0, tsq with t ą 0 are given, define @ ps, Dq P r0, ts ˆD, Lrf b Fsps, Dq B s f psqFpDq `f psqLrFspDq (124)

A simple computation shows that the process pM f bF s q sPr0,ts given by @ s P r0, ts, M f bF s f psqFpD s q ´f p0qFpD 0 q ´ż s 0 Lrf b Fspu, D u q du is a martingale, whose bracket process is given by @ s P r0, ts, @ M f bF ¨Ds " ż s 0 f 2 puqΓ L rF, FspD u q du By traditional approximations, these considerations can be generalized to more general mappings F : r0, ts ˆD Ñ R, in particular they must be C 1 with respect to the time component so that (124) can be extended to @ ps, Dq P r0, ts ˆD, LrFsps, Dq B s Fps, Dq `LrFps, ¨qspDq

The fact that the corresponding process defined by @ s P r0, ts, M F s Fps, D s q ´Fp0, D 0 q ´ż s 0 LrFspu, D u q du is a martingale is called the Dynkin's formula. Fix g P C 8 pV q, the above considerations can be applied to the mapping F : r0, ts ˆD Q ps, Dq Þ Ñ ΛrP t´s rgsspDq for which we compute LrFs " 0, due to the intertwining relation of Theorem 3. Taking expectations, it follows that E tx 0 u rΛrgspD t qs " ΛrP t rgssptx 0 uq which amounts to intertwining relations at the level of semi-groups: @ g P C 8 pV q, P t rΛrgssptx 0 uq " ΛrP t rgssptx 0 uq " P t rgspx 0 q where pP t q tě0 is the Markov semi-group associated to L. Since both the l.h.s. and the r.h.s. can be seen as integration of the mapping g, this relation is extended to any non-negative measurable function g. When we take for g the indicator function of a measurable set negligible with respect to µ, we get @ t ą 0, E tx 0 u rΛrgspD t qs " 0 according to Lemma 64 and due to the fact that µpD t q ą 0, from Theorem 5. We deduce that P t rgspx 0 q " 0, for any t ą 0 and x 0 P V , as wanted. An immediate extension is:

Proposition 65 Assume that there exists ą 0 such that for any x 0 P V , we can construct a solution pD t q tPr0, s to the martingale problem associated to pD, Lq and starting from the singleton tx 0 u Ă V . Then for any t ą 0 and whatever the initial law LpX 0 q, the law of X t is absolutely continuous with respect to µ.

Proof

The arguments presented above the statement of this proposition show that for any s P p0, s and any function f : V Ñ R `negligible with respect to µ, we have that P s rf s " 0. By invariance of µ, we also have that for any u ě 0, P u rf s is negligible with respect to µ: µrP u rf ss " µrf s " 0. We deduce that P s`u rf s " P s rP u rf ss " 0 and the announced result follows.

Of course Corollary 7 and Proposition 65 are well-known in the present elliptic diffusion framework. Nevertheless, we think this new approach can be adapted to more complicated context, as Theorem 5 is quite universal (it was shown to hold also for hypoelliptic diffusions, for the moment in dimension 1, in [START_REF] Miclo | Duality and hypoellipticity: one-dimensional case studies[END_REF]). We believe it should always be possible to associate to a diffusion some evolving sets (as mentioned in the introduction) whose weights for an invariant measure behave like a continuous martingale. By conditioning the primal diffusion X to remain inside these sets, we would be led to a Bessel-3 process, up to a time-change and at least if the randomness of X is sufficient, as the Brownian motion conditioned to stay positive ends up being a Bessel-3 process.

Another noticeable downside of Corollary 7 is that it requires the a priori knowledge that µ is absolutely continuous with respect to the Riemannian measure. A more general statement would only conclude, at positive times, to the absolute continuity of the time-marginal laws with respect to the invariante measure. In this paper we only considered kernels Λ which are directly related to the invariant measure µ, but it would be instructive to condition with respect to other measures, even time-dependent ones.

A About product situations

As already mentioned in the introduction, there are in general several dual generators intertwined through Λ with a given generator L. We consider in this appendix the product situation, where this multiplicity is particularly obvious.

Let r L and p L be two smooth generators on the manifolds r V and p V of dimension larger or equal to 1. Consider V r V ˆp V endowed with L r L b p I `r I b p L ( r I and p I are the identity operators acting on C 8 p r V q and C 8 p p V q respectively). All the notions relative to r L (respectively p L) will receive a tilde (resp. a hat). Assume that r L admits an invariant Radon measure r µ and consider on r G, an appropriate set of compact subsets of r V with positive measures, the kernel r Λ naturally associated with r µ. Let r D be an algebra of observables on r G on which we are given an operator r L, intertwined with r L through r Λ: D t q tPr0,r τ q and p p D t q tPr0,p τ q are independent processes satisfying the martingale problems associated with p r D, r Lq and p p D, p Lq respectively, then pD t q tPr0,τ q , defined by τ r τ ^p τ @ t P r0, τ q, D t p r D t , p D t q P G indep is a solution to the martingale problem associated with pD indep , L indep q.

It should be clear that such a solution is very different from the one obtained from Theorem 4, due to the fact that the evolutions on r G and p G are independent. In fact, the state spaces G indep and D are even disjoint. Consider the example where r L " p L is the Laplacian on R and add the singletons to G and D. Starting from a singleton, the solution associated with L indep evolves as rectangles (centered at the initial point) with independent side-lengths behaving as Bessel processes of dimension 3, while the solution associated with Theorem 4 evolves as disks (centered at the initial point) whose radius are Bessel process of dimension 4 (according to Subsection 2.1). It could be objected that this argument is not really valid, since we did not show uniqueness of the solution to the martingale problem associated with pD, Lq, or with formal extensions of pD, Lq, in the sense that exactly the same definitions are applied to more general subsets than those from D. But in Proposition 61, it is proven that a solution to such a martingale problem, which is furthermore defined for all times, ends up looking like a big disk and this is not true for the processes associated with pD indep , L indep q, since starting from a rectangle, it remains in the set of rectangles.

The fact that under L the evolutions of different parts of the boundary of a domain are strongly correlated could suggest to try to couple the evolutions under r L and p L. More precisely, assume that r G " r D and that r D and r L are constructed as in the introduction, similarly for p p G, p D, p Lq. Let p r D t q tPr0,r τq and p p D t q tPr0,p τq be solutions to the corresponding martingale problems. According to Proposition 53, there exist Brownian motions p r B t q tě0 and p p B t q tě0 such that @ r f P C 8 p r V q, @ t P r0, r τ q, dF r f p r D t q " r LrF In the previous independent framework, p r B t q tě0 and p p B t q tě0 are independent and we end up with the generator L indep . Now we would like to couple p r D t q tPr0,r τq with p p D t q tPr0,p τq by taking p r B t q tě0 " p p B t q tě0 , since this is suggested by a naive extension of the radial evolution (3) to the domains belonging to G indep . But again we end up with a process different from the one obtained from Theorem 4, for the same reason as above: in the case r L " p L " B 2 , it will evolve as squares if it is started from a square. It can also be seen on the action of the generators on observables of the form F where r 1 P C 8 p r V q and p 1 P C 8 p p V q are the functions always taking the value 1. But in both cases, we have the same carré du champs: for any r f P C 8 p r V q and p f P C 8 p p V q,

Γ L rF r f b p f s " Γ L equal rF r f b p f s " ´F r f b G p f `G r f b F p f ¯2 which is different from Γ L indep rF r f b p f s " F 2 r f b G 2 p f `G2 r f b F 2 p f
Nevertheless, the generator L equal is not intertwined with L through Λ. Indeed, for any r f P C 8 p r V q and p f P

C 8 p p V q, denote R r f b p f G r f b G p f ´Gr 1 F r 1 F r f b G p f ´G r f b G p 1 F p 1
F 1 L equal rΛrf ss " L equal rF f s ´2 F 1 Γ L equal rF f , F 1 s `Ff ˆ2 F 2 1 Γ L equal rF 1 , F 1 s ´1 F 1 L equal rF 1 s " L equal rF f s ´2 F 1 Γ L rF f , F 1 s `Ff ˆ2 F 2 1 Γ L rF 1 , F 1 s ´1 F 1 L equal rF 1 s " F 1 LrΛrf ss `2R f ´2 F f F 1 R 1 " F 1 ΛrLrf ss `2R f ´2 F f F 1 R 1
Thus if the generator L equal was to be intertwined with L through Λ, we would have for any r f P C 8 p r V q and p f P C 8 p p

V q, R r f b p f " F r f b p f F 1 R 1 " ´F r f b p f F 1 G

Figure 1 :

 1 Figure 1: convergence in the Hausdorff topology, not in the strong sense

Remark 12 A

 12 natural converse question is: given D, r D P D and a linear mapping T from C 8 pCq to C 8 p r Cq (with r C B r D), is there a smooth function Φ on D with ΦpDq " r D and such that T " T D Φ? The investigation of this kind of general issues is out of the scope of the present paper. Nevertheless, a first step in this direction is as follows. Let α, r α be given in T D D and T r D D respectively. Remark 16 shows how to extend α and r

1 0d 3 1 0` 1 0` 1 0` 1 0| 1 0|

 1311111 Let σ P r0, 1s and ζ σ pt, xq " σpu 0 pt, xq, ∇u 0 pt, xq, ∇∇u 0 pt, xqq, by definition of u 0 , there exists a constant C 1 ą 0 such that |ζ σ pt, xq ´ζσ ps, xq| ď C 1 |t ´s|, |ζ σ pt, xq ´ζσ pt, yq| ď C 1 |x ´y| α . Let u 0 pt, xq pu 0 pt, xq, ∇u 0 pt, xq, ∇∇u 0 pt, xqq, we have: Spu 0 qpt, xq " Φpt, x, 0, 0, 0q ´Φpt, x, u 0 pt, xq, ∇u 0 pt, xq, ∇∇u 0 pt, xqq " ´ż Φpt, x, ζ σ pt, xqqp u 0 pt, xqq dσ hence |Spu 0 qpt, xq ´Spu 0 qps, xq| " ˇˇż d3 Φpt, x, ζ σ pt, xqqp u 0 pt, xqq ´d3 Φps, x, ζ σ ps, xqqp u 0 ps, xqq ˘dσ ˇ" ˇˇż d3 Φpt, x, ζ σ pt, xqq ´d3 Φps, x, ζ σ ps, xqq ˘p u 0 pt, xqq dσ `ż d3 Φps, x, ζ σ ps, xqq ˘p u 0 pt, xq ´ u 0 ps, xqq dσ ˇď ż `d3 Φpt, x, ζ σ pt, xqq ´d3 Φps, x, ζ σ ps, xqq ˘p u 0 pt, xqq| dσ `ż `d3 Φps, x, ζ σ ps, xqq ˘p u 0 pt, xq ´ u 0 ps, xqq| dσ.

  Φps, x, ζ σ ps, xqq| ď |d 3 Φpt, x, ζ σ pt, xqq ´d3 Φpt, x, ζ σ ps, xqq| `|d 3 Φpt, x, ζ σ ps, xqq ´d3 Φps, x, ζ σ ps, xqq| ď C 1 |ζ σ pt, xq ´ζσ ps, xq| `C2 |t ´s| α{2 ď pC 1 δ 1´α{2 `C2 q|t ´s| α{2 . where C is a constant whose value can change from one line to the other (also below). Also we have | u 0 pt, xq| ď Ct ď Cδ. On the other hand we have: |d 3 Φps, x, ζ σ ps, xq| ď C and | u 0 pt, xq ´ u 0 ps, xq| ď C|t ´s|. Putting all things together we get: |Spu 0 qpt, xq ´Spu 0 qps, xq| ď Cpδq|t ´s| α{2with Cpδq tending to 0 as δ tends to 0.Let us show that: |Spu 0 qpt, xq ´Spu 0 qpt, yq| ď Cpδq|x ´y| α

  Φpt, y, ζ σ pt, yqq| ď |d 3 Φpt, x, ζ σ pt, xqq ´d3 Φpt, y, ζ σ pt, xqq| `|d 3 Φpt, y, ζ σ pt, xqq ´d3 Φpt, y, ζ σ pt, yqq| ď C 2 |x ´y| α `C1 |ζ σ pt, xq ´ζσ pt, yq| ď pC 1 `C2 q|x ´y| α , as well as | u 0 pt, xq| ď Cδ |d 3 Φpt, y, ζ σ pt, yqq| ď C Moreover |u 0 pt, xq ´u0 pt, yq| ď ż t 0

Figure 2 :

 2 Figure 2: example of a non injective mapping ψ C,r

  For any D P D and x P C, consider Ȓ`p xq 1 2 inf " r ą 0 : exp x prν C pxqq P C and ν C pexp x prν C pxqqq " ´d dr exp x prν C pxqq * Ȓ`p Dq inft Ȓ`p xq : x P Cu Similarly, let Ȓ´p xq 1 2 sup " r ă 0 : exp x prν C pxqqq P C and ν C pexp x prν C pxqqq " d dr exp x prν C pxqq * Ȓ´p Dq supt Ȓ´p xq : x P Cu

ż r 0 sinh n´1 puq du ď sinh n prq pn ´1q coshprq namely sinh n´1 prq Jprq ě pn ´1q cothprq and 2 sinh n´1 prq Jprq ´pn ´1q cothprq ě sinh n´1 prq Jprq ě 0

 00

pnq

  ΨpC,rq pψ C,r pxqq| r"0

B

  r H f pΨpD, rqq| r"0 " ż C ρ xν, ∇f y `ρp1q f `ρ2 f dσ It follows that B 3 r F f pDq| r"0 " ż C f `ρ xν, ∇f y `ρp1q f `ρ2 f dσ Proof The domain D P D being fixed, consider a tubular neighborhood T of D such that for any y P T , there exists a unique r P R and x P C such that y " ψ C,r pxq. Consider then the mapping r ρ : T Ñ R given by r ρpyq " ρ ΨpC,rq pyq. With this definition, we have for r sufficiently small, H f pΨpD, rqq " G r ρf pΨpD, rqq. It follows that B r H f pΨpD, rqq| r"0 " B r G r ρf pΨpD, rqq " ż C xν, ∇pr ρf qy `ρr ρf dσ It remains to note that on C, we have xν, ∇pr ρf qy " r ρ xν, ∇f y `f xν, ∇r ρy " ρ xν, ∇f y `f ρ p1q to get the first identity. The second one comes from the rewriting, in our present context, of the second equality in Lemma 10 as B 2 r F f pΨpD, rqq " ż ΨpD,rq f dλ `żΨpC,rq ρf dσ " F f pΨpD, rqq `Hf pΨpD, rqq

D

  r L r Λ " r Λ r L. Make similar hypotheses for p L. Next define G indep t r D ˆp D : r D P r G and p D P p Gu D indep r are the identity operators on r D and p D respectively. It is immediate to check that L indep Λ " ΛL, where Λ r Λ b p Λ is the natural Markov kernel associated with the measure µ r µ b p µ, invariant for L. When p r

  r

  r f b p f , where r f P C 8 p r V q and p f P C 8 p p V q. In the general setting, Itô's formula leads for the above coupling to the generator L equal acting on G indep as of Subsection 6.3. But simple computations show that the formal extension of L to G indep should be given by

r f b p f and 1 r 1 b p 1 ,

 1 From the proof of Theorem 3, we have, with f

r 1 b G p 1 r f b p f p r D ˆp Dq " ´r µp r Dqp µp p Dq r µpB r Dqp µpB p Dq R r f b p f p r D ˆp Dq The sets r D and p D being fixed, the mapping r f b p f Þ Ñ R r f b p f p r D ˆp Dq corresponds to an integration of r f b p f on the boundary of r D b p D, while r f b p f Þ Ñ F r f b p f p r D ˆp

 1ˆp This equality holds on G indep , namely for any r D P r D and p D P p D, we have F Dq correspond to an integration of r f b p f on the interior of r D b p D. Thus for any function r f (respectively p f ) whose support is included in the interior of r D (resp. p D), we get F r f b p f p r D ˆp Dq " 0, i.e. r µ b p µ vanishes on the interior of r D b p D. Since this is true for any r D P r D and p D P p D, we would conclude that r µ " 0 and p µ " 0, a contradiction.

  So that the last r.h.s. admits the more readable expression ż

		x∇f, νy dµ	`żΨpC,rq	f px∇U, νy `ρq dµ
		ΨpC,rq
	Proof
	The first differentiation is a classical result. It can also be deduced from the disintegration of µ with
	respect to (30) and (31). For instance for r P r0, R `pDqq, we have
		F f pΨpD, rqq " F f pDq	`ż r	ż	f dµ ds
					0	ΨpC,sq
	and the r.h.s. is easily differentiated with respect to r.
	For the second differentiation, first write
		ż	ż
		f dµ "	f exppU q dσ
		ΨpC,rq	ΨpC,rq
	To differentiate with respect to r the r.h.s., one has to adapt the arguments of Section 1.2 of the book
	of Mantegazza [17], to get
	ż	ż
	B r	f exppU q dσ "	x∇pf exppU qq, νy dσ	`żΨpC,rq	f exppU qρ dσ
	ΨpC,rq	ΨpC,rq
		ż
		"	x∇f, νy dµ	`żΨpC,rq	px∇U, νy `ρqf dµ
		ΨpC,rq

  and p is not conjugate to q then

	2pd ´1q ? k	`1 ´cosp ? kdpp, qqq sinp ? kdpp, qqq ď ρ Ĉ pqq ´ρC ppq
	(ii) if k ď 0, and p is not conjugate to q then
	2pd ´1q a	|k|	`1 ´coshp a |k|dpp, qqq sinhp a |k|dpp, qqq ď ρ Ĉ pqq ´ρC ppq

  ¨q, ∇f g pt, ¨qq , rptqq " tF 0 pxq `f g pt, xqν 0 pxq `rptqνpt, x, f g pt, ¨q, ∇f g pt, ¨qq : x P M u,In the above formula, dµ M is a Riemannian measure for a fixed metric in M and d x F g ψ pt, x, `gptq, f g pt, xq, ∇f g pt, xq, ∇∇f g pt, xqq, where V is a function regular in the four last components. It follows that there exists a constant C ą 0 such thatxpt, xq Þ Ñ d g pt, xqy α{2 ď Cp} ? 2B . } C α{2 `}g} C α{2 `}f g } C 1`α{2,2`α qWe deduce there exists C δ 0 ,δ 1 , depending on δ 0 , δ 1 and on the random quantity } ?2B . } C α{2 , such that xt Þ Ñ d g pt, xqy α{2 ď C δ 0 ,δ 1 and thus }t Þ Ñ d g pt, xq} C α{2 r0,t 0 s ď C δ 1 ,δ 0 pt

						? 2B t	gptqq
	is evaluated in an orthonormal basis for this metric. Let
	d g pt, xq detrν F g ψ pt, xq, d x F g ψ pt, x, 2B t with the semi-norm ? 2B t `gptqqs V px, ?
	xpt, xq Þ Ñ d g pt, xqy α{2	sup	"	|t ´s| α , s ‰ t P r0, t 0 s, x P M |f ptq ´f psq|	*
						α{2 0	`1q	`K
						α{2 0	`1q `KqµpM q,	(87)
	Using Stoke's Theorem we have that the volume of µpΨpG g t , ΨpBG g t , ? 2B t `gptqq is	? 2B t `gptqqq enclosed by the hypersurface
	µpΨpG g t ,
	then we have
	ΨpBG g t and	hpΨpG g t ,	? 2B t `gptqqq " 2	µpΨpBG g t , µpΨpG g t , ? ? 2B t `gptqqq 2B t `gptqqq	.

We have the following formula for the n-volume of the boundary:

µpΨpBG g t , ? 2B t `gptqqq " ş F g ψ pt,M, ? 2Bt`gptqq dµ F g ψ pt,M, ? 2Bt`gptqq " ş M detrν F g ψ pt, xq, d x F g ψ pt,

x, ? 2B t `gptqqs dµ M . (86) with K " }d g p0, .q} 8 not depending on g. Hence t Þ Ñ µpΨpBG g t , ? 2B t `gptqqq is in C α{2 and }t Þ Ñ µpΨpBG g t , ? 2B t `gptqqq} C α{2 ď pC δ 1 ,δ 0 pt

  as long as the domain ΨpG g t ,

				? 2B t	gptqq
	keeps a positive mass, which may lead us to replace t 0 by a smaller value, and we deduce that
		}t Þ Ñ hpΨpG g t ,	? 2B t `gptqqq} C α{2 pr0,t 0 sq ď C	(90)
	for a constant C that only depends on δ 0 ,δ 1 , t 0 and } ? 2B . } C α{2 . So Γpgq P C 1`α{2 . We have for
	0 ă s, t ă t 1 ď t 0	1´α{2 |Γpgqptq ´Γpgqpsq| ď |t ´s|C ď Ct 1

  `g1 ptq, f g 1 pt, xq, ∇f g 1 pt, xq, ∇∇f g 1 pt, xqq ´V px, ?2B t `g2 ptq, f g 2 pt, xq, ∇f g 2 pt, xq, ∇∇f g 2 pt, xqq µ M pdxq.We want to control the norm of the above function in C α{2 . Since it vanishes at time 0, we have only to control its semi-norm x¨y α{2 . We write for simplicity f g pt, xq pf g pt, xq, ∇f g pt, xq, ∇∇f g pt, xqq, and let

	"	1 t , M V px, ş	? 2B t `g1 ptqqq ´µpΨpBG g 2 t , ? 2B t (91) ? 2B t `g2 ptqqq
	Jpt, xq	V px,	? 2B t `g1 ptq, f g 1 pt, xqq ´V px,	? 2B t `g2 ptq, f g 2 pt, xqq
	Let σ P r0, 1s and
	ζ σ pt, xq	σp	? 2B

t `g1 ptq, f g 1 pt, xqq `p1 ´σqp ? 2B t `g2 ptq, f g 2 pt, xqq we have, for all 0 ď s, t ď t 1 , |ζ σ pt, xq ´ζσ ps, xq| ď |t ´s| α{2 p2 ?

  dpF θ pt, xq, F θ pt, yqq, and the mappings t Þ Ñ F θ pt, xq are uniformly Lipschitz on any compact r0, T s Ă r0, τq, we deduce that At a differentiability time t P r0, T s, we have, as in the proof of Proposition 30,

	d dt	px,yqPBG 2 t sup	dpx, yq
	"	d dt	pxt,ytqPBG 2 t : dpxt,ytq"diampBGtq sup	dpx t , y t q
	"	sup t : dpxt,ytq"diampBGtq pxt,ytqPBG 2	d dt	dpx t , y t q
	"	sup t : dpxt,ytq"diampBGtq pxt,ytqPBG 2	x	d dt	x t , ν BGt px t qy	`x d dt	y t , ν BGt py t qy
	"	sup t : dpxt,ytq"diampBGtq pxt,ytqPBG 2 ´ρΨpBGt,	?	2Bt`θtq pψ BGt, ? 2Bt`θt px t qq ´ρΨpBGt,	?	2Bt`θtq pψ BGt, ? 2Bt`θt py t qq
	"			sup
			pxt,ytqPBD 2
								dpx, yq " sup
							2 t	x,yPM 2
								t Þ Ñ	sup
								px,yqPBG 2

t dpx, yq is Lipschitz on r0, T s, hence almost everywhere differentiable on r0, T s and absolutely continuous. t : dpxt,ytq"diampBDtq

  According to Lemma 10 and (56), we have for any f P C 8 pR n q, dF f pD t q " dF f pΨpG t ,

	where the latter is well-defined since Bes	pn`2q 2t	ą	? 2B t for all t ą 0. It appears that
	@ t ě 0,	Bes	pn`2q 2t	"	? 2B t	`ż t	hpD s q ´ρBDs ds
								0
	hence
	@ x P B r G t ,	B t x " phpD t q ´ρΨpBGt,	? 2Btq qν BGt pxq
					" r α BGt,	? 2Bt pxqν BGt pxq
					? 2B t qq
	"	ˆżBDt	f phpD t q ´ρBDt q dµ ˙dt	`ˆż	f dµ	˙p? 2dB t q
								BDt
		`ˆż
			pn`2q 2t	"	? 2B t	`ż 2t 0	n s `1 2Bes pn`2q	p0q	ds
						"	? 2B t	`ż t 0	n Bes 2s `1 p0q pn`2q	ds
	Consider for any t ě 0,
			D t r G t		Bpx 0 , Bes Ψ ´1pD t , ? pn`2q 2t 2B t q q

BDt x∇f, ν BDt y dµ `żBDt f ρ BDt dµ ˙dt " ˆżBDt x∇f, ν BDt y `f hpD t qq dµ ˙dt `?2 ˆżBDt f dµ ˙dB t " LrF f spD t q dt `dM t where pM t q tě0

  ΨpC,rq pψ C,r pxqq| r"0

								17, which lead to
	dF f pΨpG t ,	? 2B t `θt `rqq
	"	´ˆż	BΨpGt, ? 2Bt`θt`rq	ρ BΨpGt, ? 2Bt`θtq ˝ψBGt, ?	2Bt`θt	˝ψ´1 BGt,	? 2Bt`θt`r f dσ ˙dt
								˙p?
		`ˆż	BΨpGt, ? 2Bt`θt`rq	f dσ	2dB t `Bt θ t dtq
		`ˆż BΨpGt, ? 2Bt`θt`rq	xν, ∇f y dσ	`żBΨpGt, ? 2Bt`θt`rq	ρf dσ ˙dt
	"	´ˆż	BΨpGt, ? 2Bt`θt`rq	ρ BΨpGt, ? 2Bt`θtq ˝ψBΨpGt,	?	2Bt`θt`rq,´r f dσ ˙dt
		`ˆż	BΨpGt, ? 2Bt`θt`rq	xν, ∇f y `ρf `Bt θ t f dσ ˙dt
		`?2	ˆżBΨpGt, ? 2Bt`θt`rq	f dσ ˙dB t
	"	ˆżBΨpGt, ? 2Bt`θt`rq	xν, ∇f y `´hpΨpG t ,	? 2B t `θt qq	`αp´rq t BD prq	¯f dσ ˙dt
		`?2	ˆżBΨpGt, ? 2Bt`θt`rq	f dσ ˙dB t
	" L prq rF f spD	prq t q dt `?2	˜żBD prq	f dσ ¸dB t
								t
	For any D P D, define
			@ x P C,			ρ	p1q C pxq	B r ρ

  ProofConsider the evolution described in Lemma 56. Certain terms are very easy to differentiate with respect to r: according to the first part of Lemma 10For the Brownian part, use the second part of Lemma 10:For the remaining term, we decompose the derivative in B r L prq rF f spD prq t q| r"0 " pB r L prq | r"0 qrF f spD t q `Br LrF f spD Use (110) for both terms of the r.h.s. For the first one, we get for any D P D,

							ż	ż
		@ t P r0, τ s,		B r	BD	t prq	f dσ "	Ct	x∇f, νy `ρf dσ
									prq t q| r"0
	pB r L prq | r"0 qrF f spDq "		´2ş ρ dσ λpDq ş C f dσ	`2 σpCq 2 ş C f dσ λpDq 2	`żC	ρ p1q f dσ
	For the second one, again for any D P D, taking into account that α C " 0, we have p0q
	B r LrF f spΨpD, rqq| r"0 " B r	ż ΨpD,rq		f dλ	ş ΨpC,rq f dσ λpΨpD, rqq `2 σpΨpC, rqq	ˇˇˇˇr	"0
		"	ż C	f dσ	`2ş ρ dσ λpDq ş C f dσ	´2 σpCq 2 ş C f dσ λpDq 2	`2 σpCq	ş C xν, ∇f y `ρf dσ λpDq
	Putting together these computations, we obtain
	B r L prq rF f spD t q| r"0 " prq	ż	C	f `ρp1q f dσ	λpDq `2 σpCq	ż
									t 0 ˆżCs	x∇f, νy `ρf dσ ˙dB s
	where
	@ D P D,	LrG f spDq		ż C	f	`2 σpCq λpDq	xν, ∇f y `ˆ2	σpCq λpDq	ρ `ρp1q ˙f dσ

@ t P r0, τ s, B r F f rD prq t s| r"0 " G f rD t s C xν, ∇f y `ρf dσ

which leads to the definition of LrG f s.

  ). When C is the smooth boundary of a convex domain, this can be obtained by letting r go to `8 in that for large r ą 0, ΨpC, rq is quite close to a circle of radius r.It would be interesting to see if this argument could be adapted to treat the general case. (b) Consider the Euclidean space (or any null curvature space) of dimension larger than 2. From Remark (49), we deduce that

	As in the proof of Lemma 58, we deduce that
	B r	ż	κ dσ ˇˇˇr	"	ż	κ p1q `ρκ dσ	(114)
		ΨpC,rq		"0		C
		ż				ż
				ρ dσ "		ρ dσ
		C			ΨpC,rq
	and by remarking ρ p1q pxq "	´ÿ mP n´1	λ 2 m pxq
	More generally, when V has a constant sectional curvature K, we get
	ρ p1q pxq " ´Kpn ´1q	´ÿ mP n´1	λ 2 m,C pxq
	Recall that the Gauss curvature at x P C is given by
						ź
		κ C pxq "		λ m,C pxq
					mP n´1
	Similarly to (111), we can introduce
	@ x P C,	κ	p1q C pxq	B r κ ΨpC,rq pψ C,r pxqq| r"0
	and, if one has indexed in a coherent (e.g. nondecreasing) way the eigenvalues of the second fundamental
	form,
	@ x P C, @ m P n ´1 ,		λ p1q m,C pxq	B r λ m,ΨpC,rq pψ C,r pxqq| r"0
	Then we have, at least if none of the eigenvalues vanishes,
	@ x P C,	κ	p1q C pxq " κ C pxq	ÿ mP n´1	p1q λ m,C λ m,C	pxq

  OpλpD t q 2 q where θ ´1 stands for the inverse mapping of θ : R `Ñ R `. not been able to show it, even taking into account a lower bound on the rate of escape for the Bessel process pR t q tě0 of dimension 3, stating that for any a ą 1,

					θt	"
					σpC s q dB s
					0
	r O ¨?t	`dż θs	σpC s q 2 ds	'
					0
	" r Op	? tq
	It follows that
			t 2 " r Opθ ´1 t q
				" r OpR 2 θ ´1 t	q
				" r
					Finally we obtain
			? t " r Op	a λpD t qq	(123)
	and this is sufficient to insure that a.s.
		lim tÑ`8	M t λpD t q a	" 0
	in view of (120) and (121).
	Remark 63 From (123), it appears that
		lim sup tÑ`8	lnptq lnpλpD t qq	ď 1
	We believe (in accordance with the beginning of Lemma 62) that
		lim tÑ`8	lnpλpD t qq lnptq	" 1
	but we have lim inf tÑ`8	R t ln a ptq ? t	"	`8

 ? 2 pB . q a.s. But the supplementary term hpD t q in Proposition 42 should prevent this collapsing in finite time.

Acknowledgments:

We are grateful to Marc Arnaudon, Jérôme Bertrand and Yuxin Ge for all our discussions about Riemannian geometry, to the referee for her/his suggestions and to the hospitality of the Institut Élie Cartan de Lorraine and of the Institut Mittag-Leffler, where this work was partially done.

* LM acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program, grants ANR-12-BS01-0019 and ANR-17-EURE-0010.

Proof

Since C and Ĉ are compact, and Ĉ X C " H we have δ " dpC, Ĉq ą 0. Using the continuity of the solution of (58), we get the existence of 0 ă T C ă τ C (resp. 0 ă T Ĉ ă τ Ĉ ) such that for all t P r0, T C s, we have dpC, BG t q ď δ 4 (resp. for all t ď T Ĉ we have dp Ĉ, B Ĝt q ď δ 4 ). Take τ C, Ĉ " T C ^T Ĉ .

Consider the following stochastic mean curvature evolution starting from C 0 " BD 0 dx " ´?2dB t ´ρCt pxqdt ¯νCt pxq (76)

According to the Doss and Sussman approach, a solution of ( 76) is given by pΨpG t , ? 2B t qq tPr0,τ q where pBG t q tPr0,τ q is a solution of (58) with rptq " ? 2B t . Equation ( 76) is a particular case of equation ( 75) with b " 0.

Corollary 28 Let D, D P D with C 5`α boundaries, α P p0, 1q, and C " BD, Ĉ " B D. Suppose that D Ă D, Ĉ X C " H Let pBD t q tPr0,τ C q (resp. pB Dt q tPr0,τ Ĉ q ) be a solution of (76) started at C (resp. Ĉ) then for a positive stopping time τ C, Ĉ ą 0 we have:

Proof

Use Corollary 27 we get there exist τ C, Ĉ ą 0 such that @ t P r0, τ C, Ĉ q, BG t X B Ĝt " H

We have BD t " ΨpBG t , ? 2B t q for t P r0, τ C q (resp. B Dt " ΨpB Ĝt , ? 2B t q for t P r0, τ Ĉ q). For t P r0, τ C, Ĉ q, Ψp., ? 2B t q is a diffeomorphism between BG t and its image BD t (resp. between B Ĝt and its image B Dt ). The proof of the corollary will be done by contradiction, suppose that there exists a time 0 ă t ă τ C, Ĉ such that ΨpBG t , ? 2B t q X ΨpB Ĝt , ? 2B t q ‰ H. Then there exist x P G t and x P Ĝt such that Ψ BGt px, ? 2B t q " Ψ B Ĝt px, ? 2B t q. We have and we get a contradiction.

We want to control the distance between to different hypersurface evolving by the stochastic mean curvature by quantities that only depend on the ambient curvature.

So Lemma 13 leads to the announced result.

Remark 49 Lemma 48 is only valid in dimension 2. If R 2 is replaced by R n , with n ą 2, recall that the mean curvature ρpxq at a point x from C BD, where D is a non-empty, open, bounded, connected domain with smooth boundary, is given by λ 1,C pxq `¨¨¨`λ n´1,C pxq (with the notation introduced in Lemma 45). Extending in the natural way the previous notions, it appears that @ x P C, @ m P n ´1 , λ m,ΨpC,rq pψ C,r pxqq " λ m,C pxq 1 `rλ m,C pxq (as long as r P R is such that min xPC 1 `rλ 1,C pxq ą 0). Thus to recover the mean curvature vector through the tangent mapping of Ψp¨, rq, one must consider the vector α above D given by

λ m,C pxq 1 ´rλ m,C pxq (as long as r P R is such that min xPC 1 ´rλ n´1,C pxq ą 0). Lemma 50 Assume that V is a surface of constant curvature K, D P D and r P pR ´, R `q then we have:

Proof

We only give the proof when K ą 0, the case K ă 0 can be deduced by similar computations. For x P C, let pγ x psqq s be a curve parametrized by its arc length with values in C and γ x p0q " x. Denote τ psq 9 γ x psq its unitary tangent vectors. Consider for any t, s, γps, tq exp γxpsq ptνpγ x psqqq J s ptq B s pγps, tqq

As a variation of a geodesic (for all the following Riemannian geometry notions, see e.g. the book of Gallot, Hulin and Lafontaine [START_REF] Gallot | Riemannian geometry. Universitext[END_REF]), pJ s ptqq t is a Jacobi field. We have J s p0q " τ psq and 9 J s p0q " ∇ Bs νpγ x psqq " ρ C pγ x psqqτ psq. So there exist α, β P R such that J s ptq " pα cosp ? Ktq`β sinp ? Ktqq{{ tÞ Ñγps,tq τ psq, where {{ tÞ Ñγps,tq is the parallel transport above the curve t Þ Ñ γps, tq. Adjusting with the initial condition, we get:

When the curvature is negative K ă 0, except for the sign change in the second order differential equation for the Jacobi field, all the computations are similar.

Remark 51 In the context of the above lemma, let V be a pn `1q-dimensional manifold with constant curvature K ą 0, D P D, r P pR ´, R `q and λ C,1 pxq ď ... ď λ C,n pxq be the principal curvatures of C. It is not so clear how to control the principal curvatures of ΨpC, rq at the point ψ C,r pxq, but for the mean curvature we have:

Comparison of two Doss-Sussman approaches

Consider the Doss-Sussman method corresponding to the decomposition (57) of Remark 18. Similarly to (43) and (47), define in the present Riemannian Brownian setting,

We are interested in constructing a family p r G t q tPr0,τq such that # r G 0 " Bpx, r 0 q @ t P r0, τq,

since the process pD t q tPr0,τq obtained by a particular composition of the normal flow Ψ and of the flow (95), namely @ t P r0, τq, D t Ψp r G t , ? 2B t q (96) will provide a solution to the martingale problem associated to pD, Lq, as in Theorem 17.

In the following subsections we reformulate the results of Section 2, using this Doss-Sussman approach.

: kolehe.coulibaly@univ-lorraine.fr Institut Élie Cartan de Lorraine Université de Lorraine, Site de Nancy B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex ; miclo@math.cnrs.fr Institut de Mathématiques de Toulouse Université Paul Sabatier, 118, route de Narbonne 31062 Toulouse cedex 9, France Toulouse School of Economics, 1, Esplanade de l'université 31080 Toulouse cedex 06, France