Nicolas Champagnat
email: nicolas.champagnat@inria.fr

Koléhè Abdoulaye Coulibaly-Pasquier

Denis Villemonais
email: denis.villemonais@univ-lorraine.fr

Exponential convergence to quasi-stationary distribution for multi-dimensional diffusion processes

Keywords: Diffusions on Riemannian manifolds, diffusions in a bounded domain, absorption at the boundary, quasi-stationary distribution, Q-process, uniform exponential mixing, two-sided estimates, gradient estimates. 2010 Mathematics Subject Classification. Primary: 60J60, 37A25, 60B10, 60F99. Secondary: 60J75, 60J70

We consider diffusion processes killed at the boundary of Riemannian manifolds. The aim of the paper if to provide two different sets of assumptions ensuring the exponential convergence in total variation norm of the distribution of the process conditioned not to be killed. Our first criterion makes use of two sided estimates and applies to general Markov processes. Our second criterion is based on gradient estimates for the semi-group of diffusion processes.

Introduction

We consider a Markov process X evolving in a Riemannian compact manifold (M, ρ) of dimension d ≥ 1 with boundary ∂M , such that, when it hits ∂M , X is killed and immediately sent to a cemetery point ∂ / ∈ M . We are interested in providing sufficient criteria ensuring exponential mixing for the distribution of X conditioned not to be killed when it is observed. More precisely, our goal is to prove the existence of a probability measure α on M and of positive constants C, γ > 0 such that, for all initial distribution µ on M ,

P µ (X t ∈ • | t < τ ∂ ) -α(•) T V ≤ Ce -γt , ∀t ≥ 0, (1.1)
where τ ∂ = inf{t ≥ 0, X t = ∂} is the killing time of X, P µ is the law of X with initial distribution µ and • T V is the total variation norm on finite signed measures. In the whole paper we will consider diffusions such that P x (τ ∂ < ∞) = 1 and P x (t < τ ∂ ) > 0, for all x ∈ M and all t ≥ 0. It is well known that (1.1) entails that α is the unique quasi-stationary distribution for X, that is the unique probability measure satisfying

α(•) = P α (X t ∈ • | t < τ ∂ ) , ∀t ≥ 0.
We provide two independent sets of assumptions to check the exponential mixing property of the conditional distribution of X.

Our first result (Theorem 2.1) shows that a two-sided estimate for the transition density of a general absorbed Markov process at some time t > 0 is sufficient to ensure (1.1). Two sided estimates have been recently developed for a wide range of processes, including diffusion processes in general domains of R d (we refer to the beginning of Section 2 for a bibliography). In particular, we recover the results of Knobloch and Partzsch [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF], who proved that (1.1) holds for a class of diffusion processes evolving in R d (d ≥ 3), using two sided estimates combined with non-trivial spectral properties of the infinitesimal generator of X. We actually prove that the two sided estimates are sufficient for diffusion processes in R d but also for general Markov processes, while some spectral properties can be recovered from our results.

Our second result (Theorem 3.1) is based on gradient estimates of the Dirichlet semi-group obtained by Wang [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF] and Priola and Wang [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF]. The gradient estimates of [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF] hold for Brownian motions with C 1 drift evolving in bounded manifolds with C 2 boundary ∂M and killed when they hit ∂M . The gradient estimates of [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] hold for uniformly elliptic diffusion processes with Hölder diffusion coefficient and bounded drift evolving in a bounded domain M of R d (d ≥ 1) with C 2 boundary and killed when they hit ∂M . In both situations, we prove that the convergence (1.1) holds true. Up to our knowledge, this result is the first one of this kind for diffusion processes in Riemannian manifolds and our conditions for diffusion processes on R d improve significantly the existing results of [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF] and [START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF]. In particular, we prove that the global Lipschitz and local regularity assumptions near the boundary of [START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF] are unnecessary. A fortiori the C 1 regularity assumption on the diffusion coefficient of [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF] is also strongly relaxed, when the boundary of the domain is of class C 2 and the drift is bounded and continuous (in [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF] as in Section 2, domains with C 1,1 boundary and drifts in a Kato class are allowed).

Our two approaches are complementary. The one based on two sided estimates is particularly adapted to irregular domains (see for instance [START_REF] Lierl | The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms[END_REF]), while the gradient estimate approach applies to diffusion processes with less regular coefficients.

The usual tools to prove convergence as in (1.1) involve coupling arguments: for example, contraction in total variation norm for the nonconditioned semi-group can be obtained using mirror and parallel coupling, see [START_REF] Lindvall | Coupling of multidimensional diffusions by reflection[END_REF][START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF][START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF]. However, the process conditioned not to be killed up to a given time t > 0 is a time-inhomogeneous diffusion process with a singular drift for which these methods fail. For instance, a standard d-dimensional Brownian motion (B t ) t≥0 conditioned not to exit a smooth domain D ⊂ R d up to a time t > 0 has the law of the solution (X

(t) s ) s∈[0,t] to the stochastic differential equation dX (t) s = dB s + [∇ ln P • (t -s < τ ∂ )] (X (t) s )ds.
Since P x (t -s < τ ∂ ) vanishes when x converges to the boundary ∂D of D, the drift term in the above SDE is singular and existing coupling methods do not apply. Hence, convergence of conditioned diffusion processes have been obtained up to now using (sometimes involved) spectral theoretic arguments (see for instance, [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Kolb | Spectral analysis of diffusions with jump boundary[END_REF][START_REF] Littin | Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular[END_REF][START_REF] Miura | Ultracontractivity for Markov semigroups and quasistationary distributions[END_REF] for one-dimensional diffusion processes and [START_REF] Cattiaux | Competitive or weak cooperative stochastic lotka-volterra systems conditioned to non-extinction[END_REF][START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF] for multi-dimensional diffusion processes) which are strictly limited to diffusion processes and have limitations in terms of generality and flexibility. The strength of our approach is that, instead of using a spectral theoretic perspective, we rely, as was done in [START_REF] Champagnat | Exponential convergence to quasistationary distribution for one-dimensional diffusions[END_REF] for one-dimensional diffusions, on recent probabilistic criteria for convergence of conditioned processes obtained in [4, Theorem 2.1], overcoming the difficulties pointed out in the previous paragraph. This result states that the exponential convergence (1.1) is equivalent to the following condition.

Condition (A).

There exist t 0 , c 1 , c 2 > 0 such that, for all x, y ∈ M , there exists a probability measure ν x,y on M satisfying (A1)

P x (X t 0 ∈ • | t 0 < τ ∂ ) ≥ c 1 ν x,y (•) and P y (X t 0 ∈ • | t 0 < τ ∂ ) ≥ c 1 ν x,y (•);
(A2) for all z ∈ M and all t ≥ 0,

P νx,y (t < τ ∂ ) ≥ c 2 P z (t < τ ∂ ).
Condition (A) also implies that the conditional distributions of X are in fact contracting in total variation [START_REF] Champagnat | Exponential convergence to quasistationary distribution and Q-process[END_REF]Corollary 2.2], that e λ 0 t P x (t < τ ∂ ) converges when t → +∞, uniformly in x, to a positive eigenfunction η of the infinitesimal generator of (X t , t ≥ 0) for the eigenvalue -λ 0 characterized by the relation P α (t < τ ∂ ) = e -λ 0 t , ∀t ≥ 0 [4, Proposition 2.3], a spectral gap property [START_REF] Champagnat | Exponential convergence to quasistationary distribution and Q-process[END_REF]Corollary 2.4], and the existence and exponential ergodicity of the so-called Q-process, defined as the process X conditioned to never hit the boundary [START_REF] Champagnat | Exponential convergence to quasistationary distribution and Q-process[END_REF]Theorem 3.1].

The paper is organized as follows. In Section 2, we state and prove a sufficient criterion for (1.1) based only on two-sided estimates. We also provide several references giving two sided estimates for many kinds of processes. In Section 3, we state a sufficient criterion for (1.1) for drifted Brownian motion on Riemannian manifolds and for diffusion processes with Hölder diffusion coefficients. The proof, based on gradient estimates obtained in [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF][START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF], is given in Section 4.

2 Quasi-stationary behavior under two-sided estimates

In this section, we consider absorbed Markov processes satisfying two-sided estimates. More specifically, we consider a diffusion process X in some closed Riemanian manifold M (or open subset of R d ), absorbed at the boundary of M at first hitting time τ ∂ . We assume that M is equipped with a measure µ such that for some t > 0 and for all x ∈ M , X t admits a density function p(t, x, y) in M with respect to µ(dy) given X 0 = x. We say that X satisfies a two-sided estimate at time t if there exists a constant c > 0 and two measurable function

f 1 , f 2 : M → [0, +∞) such that c -1 f 1 (x)f 2 (y) ≤ p(t, x, y) ≤ cf 1 (x)f 2 (y), ∀x, y ∈ M. (2.1)
Our main result states that this condition, together with the assumption µ(f 1 f 2 ) > 0, guaranties the uniform exponential convergence in total variation of the conditional distributions of X s given s < τ ∂ to a unique quasistationary distribution. Two-sided estimates were already a key ingredient to prove exponential convergence of conditional distributions in [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF]. However, their analysis requires additional assumptions (continuity of the transition density and existence of ground states) and they assume that the two-sided estimate holds with f 1 and f 2 the ground states of the generator and its adjoint. In addition, checking their criterion involves intricate spectral theory arguments, whereas our proof only requires to check the above two-sided estimate. Moreover, our approach leads to explicit rates of convergence in terms of c, f 1 , f 2 and µ. Also note that the existence of the ground states are recovered afterwards from our results: an eigenmeasure for the adjoint generator is given by the quasi-stationary distribution α [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF] and an eigenfunction for the generator by [START_REF] Champagnat | Exponential convergence to quasistationary distribution and Q-process[END_REF]Proposition 2.3]. Moreover, as will appear clearly in the proof, we don't use at all the Riemannian structure of M nor the diffusion property of X. In particular, our result applies to any sub-Markov process in any measurable state space M equipped with some measure µ, whose semi-group exhibits the property (2.1) for some t > 0. Finally, since our criterion implies Condition (A), we obtain many complementary results as explained in the Introduction. In particular, we obtain a contraction result in total variation (see (2.3) below), the uniform convergence of e λ 0 t P x (t < τ ∂ ) to the ground state when t → +∞ [4, Proposition Estimates of the form (2.1) have been proved in a variety of contexts. In the case of diffusion processes in R d , d ≥ 3 absorbed at the boundary of a bounded domain, this goes back to the seminal paper of Davies and Simon [START_REF] Davies | Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians[END_REF]. The case of standard Brownian motion in a bounded C 1,1 domain of R d , d ≥ 3 was studied in [START_REF] Zhang | The boundary behavior of heat kernels of Dirichlet Laplacians[END_REF]. This result has then been extended in [START_REF] Kim | Estimates on Green functions and Schrödingertype equations for non-symmetric diffusions with measure-valued drifts[END_REF] to diffusions in a bounded C 1,1 domain in R d , d ≥ 3 or more, with inifnitesimal generator

L = 1 2 d i,j=1 a ij ∂ i ∂ j + d i=1 b i ∂ i ,
with symmetric, uniformly elliptic and C 1 diffusion matrix (a ij ) 1≤i,j≤d , and with drift (b i ) 1≤i≤d in the Kato class K d,1 , which contains L p (dx) functions for p > d. Diffusions on bounded, closed Riemannian manifolds with very irregular boundary and with generator

L = ∆ + X,
where ∆ is the Laplace-Beltrami operator and X is a smooth vector field, were also studied in [START_REF] Lierl | The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms[END_REF].

Estimates of the form (2.1) are also known for quite general one-dimensional diffusions (see for instance [START_REF] Miura | Ultracontractivity for Markov semigroups and quasistationary distributions[END_REF], where the link with quasi-stationary distribution is studied).

Two-sided estimates are also known for other classes of processes, among which α-stable Lévy processes (generated by the fractional Laplacian) with drift in a Kato class in a C 1,1 domain of R d (d ≥ 2) [START_REF] Chen | Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation[END_REF], or without drift for more general domains in any dimension [START_REF] Bogdan | Heat kernel estimates for the fractional Laplacian with Dirichlet conditions[END_REF]. Extensions to more general jump processes similar to the α-stable Lévy process are given in [START_REF] Chen | Heat kernel estimates for ∆ + ∆ α/2 in C 1,1 open sets[END_REF][START_REF] Kim | Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C 1,η open sets[END_REF][START_REF] Chen | Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation[END_REF]. For example, the results of the last reference covers α-stable Lévy processes with killing (either at some state-dependent rate, of with some state-dependent probability at each jump time).

Remark 1. Note that most of these results are actually obtained for Markov processes with additional killing inside the domain, and the next result also covers this case. Indeed, two sided estimates for processes with bounded killing rate are immediate consequences of the two sided estimates for the process without killing Theorem 2.1. Assume that P x (τ ∂ < ∞) = 1 for all x ∈ M , that there exists t 0 > 0 such that X t 0 admits a density function p(t 0 , x, y) in M with respect to a probability measure µ(dy) on M given X 0 = x and that there exist a constant c > 0 and two functions

f 1 , f 2 : M → [0, +∞) such that c -1 f 1 (x)f 2 (y) ≤ p(t 0 , x, y) ≤ cf 1 (x)f 2 (y), ∀x, y ∈ M, (2.2)
and such that µ(f 1 f 2 ) > 0, then Assumption (A), and hence (1.1), are satisfied. In addition, for all µ 1 , µ 2 probability measures on M ,

P µ 1 (X t ∈ • | t < τ ∂ ) -P µ 2 (X t ∈ • | t < τ ∂ ) T V ≤ c 3 µ(f 2 ) (1 -c -5 µ(f 1 f 2 )) ⌊t/t 0 ⌋ µ 1 -µ 2 T V µ 1 (f 1 ) ∧ µ 2 (f 1 ) . (2.3)
In addition, there exists a unique quasi-stationary distribution α for X, which satisfies

c -2 f 2 (y)µ(dy) M f 2 (x)µ(dx) ≤ α(dy) ≤ c 2 f 2 (y)µ(dy) M f 2 (x)µ(dx) . (2.4)
Proof of Theorem 2.1. Let us first check that (A1) holds with ν x,y = ν independent of x, y given by

ν(dy) = f 2 (y)µ(dy) µ(f 2 ) .
Note that the assumptions of Theorem 2.1 imply that µ(f 2 ) ∈ (0, +∞), and

f 1 ∞ µ(f 2 ) ≤ c. (2.5)
In particular, ν is a well defined probability measure. For all x ∈ M and A ⊂ M , we have, by (2.2)

c -1 f 1 (x) A f 2 (y) dµ(y) ≤ P x (X t 0 ∈ A) ≤ cf 1 (x) A f 2 (y) dµ(y).
In particular,

P x (X t 0 ∈ A | t 0 < τ ∂ ) = P x (X t 0 ∈ A) P x (X t 0 ∈ E) ≥ c -2 A f 2 (y) µ(dy) µ(f 2 ) .
We thus obtained (A1) with c 1 = c -2 and ν x,y = ν.

Let us now check (A2). We have, for all z ∈ M ,

P ν (X t 0 ∈ A) ≥ c -1 M f 1 (x) dν(x) A f 2 (x) µ(dx) ≥ c -1 µ(f 1 f 2 ) µ(f 2 ) f 1 (z) f 1 ∞ A f 2 (x) µ(dx) ≥ c -3 µ(f 1 f 2 ) P z (X t 0 ∈ A),
where we used (2.5) in the last inequality.

Hence, for all t ≥ t 0 and all z ∈ M ,

P ν (t < τ ∂ ) = E ν P Xt 0 (t -t 0 < τ ∂ ) ≥ c -3 µ(f 1 f 2 ) E z P Xt 0 (t -t 0 < τ ∂ ) = c -3 µ(f 1 f 2 ) P z (t < τ ∂ ).
For t < t 0 , we have

P ν (t < τ ∂ ) ≥ P ν (t 0 < τ ∂ ) ≥ P ν (t 0 < τ ∂ )P z (t < τ ∂ ).
Integrating the first inequality in (2.2) with respect to ν(dx)µ(dy), we have

µ(f 1 f 2 ) = ν(f 1 )µ(f 2 ) ≤ cP ν (t 0 < τ ∂ ). Hence P ν (t < τ ∂ ) ≥ c -1 µ(f 1 f 2 )P z (t < τ ∂ ) for t < t 0 and we have proved (A2) for c 2 = (c -3 ∧ c -1 )µ(f 1 f 2 ) = c -3 µ(f 1 f 2 ).
We now prove (2.3). We make use of the following general consequence of Condition (A), proved in [4, Corollary 2.2]: for all probability measures µ 1 , µ 2 on M and for all t > 0,

P µ 1 (X t ∈ • | t < τ ∂ ) -P µ 2 (X t ∈ • | t < τ ∂ ) T V ≤ (1 -c 1 c 2 ) ⌊t/t 0 ⌋ c(µ 1 ) ∧ c(µ 2 ) µ 1 -µ 2 T V , (2.6)
with c(µ i ) defined by

c(µ i ) = inf t≥0 P µ i (t < τ ∂ ) sup z∈M P z (t < τ ∂ )
.

Hence we need to prove that, for i = 1, 2,

c(µ i ) ≥ c -3 µ(f 2 )µ i (f 1 ). (2.7)
We proceed as above. Assume first t ≥ t 0 . Then, using (2.2),

P µ i (X t 0 ∈ A) ≥ c -1 M f 1 (x) dµ i (x) A f 2 (x) dµ(x) ≥ c -1 µ i (f 1 ) f 1 (z) f 1 ∞ A f 2 (x) dµ(x) ≥ c -3 µ i (f 1 )µ(f 2 ) P z (X t 0 ∈ A),
where we also used (2.5). Now, for all t ≥ t 0 ,

P µ i (t < τ ∂ ) = E µ i P Xt 0 (t -t 0 < τ ∂ ) ≥ c -3 µ i (f 1 )µ(f 2 ) E z P Xt 0 (t -t 0 < τ ∂ ) = c -3 µ i (f 1 )µ(f 2 ) P z (t < τ ∂ ),
which implies (2.7) for t ≥ t 0 . For t ≤ t 0 , using (2.2),

P µ i (t < τ ∂ ) ≥ P µ i (t 0 < τ ∂ ) ≥ P µ i (t 0 < τ ∂ ) sup x∈M P x (t < τ ∂ ) ≥ c -1 µ i (f 1 )µ(f 2 ) sup x∈M P x (t < τ ∂ ) ≥ c -3 µ i (f 1 )µ(f 2 ) sup x∈M P x (t < τ ∂ ),
where the last inequality comes from the fact that c ≥ 1. This completes the proof of (2.3). The existence and uniqueness of a quasi-stationary distribution α follows easily (see [4, Thm. 2.1]).

To conclude the proof of Theorem 2.1, it only remains to check (2.4). For this, given any bounded measurable g : M → R + , since α is a quasistationary distribution,

α(g) = E α [g(X t ) | t < τ ∂ ] = M 2 g(y)p(t, x, y)α(dx)µ(dy) M 2 p(t, x, y)α(dx)µ(dy)
.

Using twice (2.2), we obtain

c -2 ν(g) ≤ α(g) ≤ c 2 ν(g).
The conclusion follows.

3 Quasi-stationary behavior under gradient estimates

In this section, we consider the two following distinct situations and prove that the exponential convergence (1.1) holds for both of them. More precisely, we assume that either S1. X is a diffusion process evolving in a bounded, connected and closed Riemannian manifold M with C 2 boundary ∂M and the infinitesimal generator of X is given by L = 1 2 ∆ + Z, where ∆ is the Laplace-Beltrami operator and Z is a C 1 vector field. The process is absorbed when it hits the boundary ∂M . for some function g such that 1 0 g(r)dr < ∞. The process is absorbed when it hits the boundary ∂M .

The reason why we concentrate on these two examples is that we will make an extensive use of gradient estimates developed by Wang in [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF] and Priola and Wang in [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] respectively for the former and the latter case. In both cases, the authors have proved that there exists t 1 > 0 and C < ∞ such that, for all f bounded measurable,

∇P t 1 f ∞ ≤ C f ∞ , (3.2)
where (P t ) denotes the semi-group of X. Both [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF] and [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] actually give a stronger version of (3.2):

∇P t f ∞ ≤ c 1 ∧ √ t f ∞ , ∀t > 0, (3.3)
with proofs relying on a careful study of some coupling for copies of X. Situation S1 corresponds exactly to the assumptions of [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF]. In Situation S2, we need to assume that M is bounded and b is bounded on M , which is stronger than what [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] assumes. Our assumptions clearly imply (i), (ii), (iv) of [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF]Hyp. 4.1] (see [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF]Lemma 3.3] for the assumption on s). Since we assume that M is bounded and C 2 , assumptions (iii') and (v) of [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] are also satisfied (see [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF]Rk. 4.2]). Note that (3.1) is satisfied as soon as s is uniformly α-Hölder on M for some α > 0.

We denote by ρ the Riemannian distance on M and by ρ ∂M (•) := inf x∈∂M ρ(x, •) the distance to ∂M . Theorem 3.1. Assume that X is a diffusion process as in situations S1 or S2 above. Then Condition (A) and hence (1.1) are satisfied. Moreover, there exist two constants C, γ > 0 such that, for any initial distributions µ 1 and µ 2 on M ,

P µ 1 (X t ∈ • | t < τ ∂ ) -P µ 2 (X t ∈ • | t < τ ∂ ) T V ≤ Ce -γt µ 1 (ρ ∂M ) ∧ µ 2 (ρ ∂M ) µ 1 -µ 2 T V . (3.4)
Before turning to the proof of this theorem, we emphasize the novelty of Theorem 3.1 in situation S1, since, up to our knowledge, no quasi-stationary distribution result has been obtain for diffusion processes evolving in such general Riemannian manifolds. As explained in the introduction, Theorem 3.1 in situation S2 is a significant improvement of [START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF] and [START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF]. In addition, compared with these two papers, the contraction inequality (3.4) is a completely new result for diffusion processes, as well as the complementary results mentioned in the introduction (uniform convergence of e λ 0 t P x (t < τ ∂ ) to the ground state when t → +∞, a spectral gap result and the uniform exponential ergodicity of the Q-process, see [START_REF] Champagnat | Exponential convergence to quasistationary distribution and Q-process[END_REF]).

Remark 2. The gradient estimates of [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] are proved for diffusion processes with space-dependent killing rate V : M → [0, ∞). More precisely, they consider infinitesimal generators of the form

L = 1 2 d i,j=1 [ss * ] ij ∂ i ∂ j + d i=1 b i ∂ i -V
with V bounded measurable. Our proof also applies to this setting, although adding a bounded killing rate would make our notations much more intricate.

4 Proof of Theorem 3.1

Since a bounded domain of R d with C 2 boundary is also a Riemannian manifold, we will keep the latter terminology in the proof. It is important to keep in mind that S1 and S2 correspond to quite different situations. However, most of the proof applies to both situations simultaneously because it mainly relies on (3.2), which has been proved in both settings, respectively in [START_REF] Wang | Gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities[END_REF] and [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF].

Estimates of the boundary's hitting probability and return time to a compact

Since the boundary ∂M is C 2 and compact, there exists ε 0 > 0 such that ρ ∂M is C 2 on M \ M ε 0 , where

M ε 0 := {x ∈ M : ρ ∂M (x) ≥ ε 0 }.
The goal of this subsection is to prove that there exists constants ε 2 ∈ (0, ε 0 ) and A > 0 such that, for all x ∈ M and all s ≥ t 1 , where t 1 comes from (3.2),

P x (X s ∈ M ε 2 | s < τ ∂ ) ≥ A. (4.1)
First, we deduce immediately from the gradient inequality (3.2) applied to f = ½ M that there exists a constant C > 0 such that, for all x ∈ M ,

P x (t 1 < τ ∂ ) ≤ C ρ ∂M (x). (4.2)
The following Lemma is proved at the end of this subsection.

Lemma 4.1. There exist ε 1 ∈ (0, ε 0 ) and a constant C ′ > 0 such that, for all x ∈ M ,

P x (T ε 1 < t 1 < τ ∂ ) ≥ C ′ ρ ∂M (x), (4.3)
where

T ε = inf{t ≥ 0, X t ∈ M ε }.
This lemma and (4.2) imply that, for all x ∈ M ,

P x (T ε 1 ≤ t 1 | t 1 < τ ∂ ) = P x (T ε 1 ≤ t 1 < τ ∂ ) P x (t 1 < τ ∂ ) ≥ C ′ C .
Since the diffusion is uniformly elliptic, it is clear using local charts that

C 2 := inf y∈Mε 1 P y (ρ(X s , y) ≤ ε 1 /2, ∀s ∈ [0, t 1 ]) > 0.
Hence, for all x ∈ M \ M ε 1 ,

P x (X t 1 ∈ M ε 1 /2 ) ≥ P x (T ε 1 < t 1 ) inf y∈∂Mε 1 P y (T ε 1 /2 > t 1 ) ≥ C ′ C 2 P x (t 1 < τ ∂ ) C For x ∈ M ε 1 , we have P x (X t 1 ∈ M ε 1 /2 ) ≥ C 2 ≥ C 2 P x (t 1 < τ ∂ ).
Finally, we deduce that there exists a constant A > 0 such that, for all x ∈ M ,

P x (X t 1 ∈ M ε 1 /2 | t 1 < τ ∂ ) ≥ A. (4.4)
This is (4.1) for s = t 1 and

ε 2 = ε 1 /2.
To conclude, for some fixed s > t 1 and x ∈ M , we deduce from the Markov property that

P x (X s ∈ M ε 1 /2 ) = E x ½ s-t 1 <τ ∂ P X s-t 1 (X t 1 ∈ M ε 1 /2 ) ≥ A E x ½ s-t 1 <τ ∂ P X s-t 1 (t 1 < τ ∂ ) = A P x (s < τ ∂ ).
Proof of Lemma 4.1. For all t < T ε 0 , we define Y t = ρ ∂M (X t ). In both situations S1 and S2, we have

dY t = σ t dB t + b t dt,
where B is a standard Brownian motion, where σ t ∈ [σ, σ] and |b t | ≤ b are adapted continuous processes, with 0 < σ, σ, b < ∞. Using Dubins-Schwarz Theorem, there exists a differentiable function c(s) such that c(0) = 0 and

W s := c(s) 0 σ t dB t is a Brownian motion and c ′ (s) ∈ [σ -2 , σ -2 ]. In addition, c(s) 0 b t dt ≥ -bc(s) ≥ -bσ -2 s. As a consequence, setting Z s = Y 0 + W s -bσ -2 s, we have almost surely Z s ≤ Y c(s) for all s such that c(s) ≤ T ε 0 .
Setting a = bσ -2 , the function

f (x) = e 2ax -1
2a is a scale function for the drifted Brownian motion Z. The diffusion process defined by N t = f (Z t ) is a martingale and its speed measure is given by s(dv) = dv (1+2av) 2 . The Green function for one-dimensional diffusion processes [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Lemma 23.10] entails, for ε = f (ε 0 ) and all u ∈ (0, ε/2) (in the following lines, P N u denotes the probability with respect to N with initial position N 0 = u),

P N u (t < T N 0 ∧ T N ε/2 ) ≤ E N u (T N 0 ∧ T N ε/2 ) t = 2 t ε/2 0 1 - u ∨ v ε/2 (u ∧ v)s(dv) ≤ 2 t ε/2 0 (u ∧ v) dv (1 + 2av) 2 ≤ u C ε t , where C ε = 2 ε/2 0 dv (1 + 2av) 2 , ( 4.5)
where we set

T N ε = inf{t ≥ 0, N t = ε}. Let us fix t 2 = εC ε . Since N is a martingale, we have, for all u ∈ (0, ε/2), u = E N u (N t 2 ∧T N ε/2 ∧T N 0 ) ≤ ε 2 P N u (T N ε/2 < t 2 ∧ T N 0 ) + ε 2 P N u (t 2 < T N ε/2 ∧ T N 0 ) ≤ ε 2 P N u (T N ε/2 < t 2 ∧ T N 0 ) + u 2 ,
where the last inequality comes from (4.5). Reducing ε 0 (and hence ε) if necessary, we can assume without loss of generality that t 2 = εC ε ≤ σ 2 t 1 .

Hence there exists a constant C > 0 only depending on ε 0 such that

P N u (T N ε/2 < σ 2 t 1 ∧ T N 0 ) ≥ C u.
Hence, setting

ε 1 = f -1 (ε/2), for all x ∈ M \ M ε 1 , P x (T Z ε 1 < σ 2 t 1 ∧ T Z 0 ) ≥ C f (ρ ∂M (x)) ≥ C ρ ∂M (x).
Now, using the fact that the derivative of the time change c s belongs to [σ, σ], it follows that for all x ∈ M \ M ε 1 ,

P x (T Y ε 1 < t 1 ∧ T Y 0 ) ≥ P x (c(T Z ε 1 ) ≤ c(σ 2 t 1 ) ∧ c(T Z 0 )) = P x (T Z ε 1 < σ 2 t 1 ∧ T Z 0 ) ≥ Cρ ∂M (x).
Therefore,

P x (T Y ε 1 ≤ t 1 ≤ T Y 0 ) ≥ E x ½ T Y ε 1 ≤t 1 ∧T Y 0 P X T Y ε 1 (t 1 < τ ∂ ) ≥ P x (T Y ε 1 ≤ t 1 ∧ T Y 0 ) inf y∈Mε 1 P y (t 1 < τ ∂ ) ≥ C ′ ρ ∂M (x),
where we used that inf y∈Mε 1 P y (t 1 < τ ∂ ) > 0. Finally, since T ε 1 = 0 under

P x for all x ∈ M ε 1 , replacing C ′ by C ′ ∧ diam(M ) -1 entails (4.
3) for all x ∈ M .

Dobrushin coefficient for the fixed time horizon conditioned distribution

For all x, y ∈ M and t > 0, let µ t x,y be the infimum measure of δ x P t and δ y P t , i.e. for all measurable A ⊂ M , µ t

x,y (A) := inf

A 1 ∪A 2 =A (δ x P t ½ A 1 + δ y P t ½ A 2 ).
For all z ∈ M and r 0 > 0, we define for all x ∈ M f z,r 0 (x) = (r 0 -ρ(x, z)) + ,

where a + = a ∨ 0. For all t > 0 and r 0 > 0, we define E t,r 0 := (x, y, z) ∈ M 3 : ∀r > r 0 , µ t x,y (f z,r ) > 0 and

E r 0 := t∈(1,2]
E t,r 0 .

Our first goal is to prove the next lemma.

Lemma 4.2. For all t > 0, (x, y) → µ t x,y (M ) is continuous on M 2 . In addition, E r 0 = M 3 for all r 0 > 0.

Proof. By (3.3), for all positive measurable f on M bounded by 1, for all x, y ∈ M and for all t ∈ (1, 2]

|P t f (x) -P t f (y)| ≤ cρ(x, y) 1 ∧ √ t ≤ cρ(x, y). (4.6)
This implies the uniform Lipschtiz-continuity of P t f for all f bounded by 1.

In particular, we deduce that

µ t x,y (M ) = inf A 1 ∪A 2 =M (P t ½ A 1 (x) + P t ½ A 2 (y))
is continuous w.r.t. (x, y) ∈ M 2 (and even Lipschitz). We now prove the second statement of Lemma 4.2. Fix r 0 > 0 and x ∈ M . Since the drift is uniformly bounded and the diffusion is uniformly elliptic, it is clear using local charts that P x (X t ∈ B(x, r 0 /2), ∀t ∈ [0, 2]) > 0.

In particular inf

t∈ [START_REF] Bogdan | Heat kernel estimates for the fractional Laplacian with Dirichlet conditions[END_REF][START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF] E x (f x,r 0 (X t )) > 0, (4.7)

and thus (x, x, x) ∈ E r 0 for all x ∈ M . Now, for all A 1 and A 2 measurable such that A 1 ∪ A 2 = M , the map (x, y, z)

→ P t (f z,r 0 ½ A 1 )(x) + P t (f z,r 0 ½ A 2 )(y)
is uniformly Lipschitz on M 3 by (4.6) and since

f z,r 0 -f z ′ ,r 0 ∞ ≤ ρ(z, z ′ ). Therefore, (x, y, z) → inf t∈(1,2] µ t x,y (f z,r 0 ) = inf t∈(1,2] inf A 1 ∪A 2 =M (P t (f z,r 0 ½ A 1 )(x)+P t (f z,r 0 ½ A 2 )(y))
is Lipschitz on M 3 . Hence, it follows from (4.7) that E r 0 contains an open ball. In particular, the interior int(E r 0 ) is non-empty. Let us prove that int(E r 0 ) is closed, which will imply by connexity that E r 0 ⊃ int(E r 0 ) = M 3 .

Let (x 0 , y 0 , z 0 ) be an accumulation point of int(E r 0 ). Fix r > r 0 . Then there exist (x, y, z) ∈ E r 0 such that ρ(z, z 0 ) ≤ (r -r 0 )/2, and r ∈ (0, (rr 0 )/2) such that the closed ball centered at (x, y, z) and of radius r is included in E r 0 . Moreover, since M is connected, the diffusion is uniformly elliptic and the drift is bounded, for all s > 0,

P x 0 (X s ∈ B(x, r)) =: c x 0
s > 0 and P y 0 (X s ∈ B(y, r)) =: c y 0 s > 0.

It then follows that, for all t 0 ∈ (1, 2], defining η = (t 0 -1)/2 ∈ (0, 1 2 ],

µ t 0 x 0 ,y 0 (f z 0 ,r ) = inf A 1 ∪A 2 =M [δ x 0 P 1+2η (f z 0 ,r ½ A 1 ) + δ y 0 P 1+2η (f z 0 ,r ½ A 2 )] ≥ c x 0 η ∧ c y 0 η × inf A 1 ∪A 2 =M inf x∈B(x,r) δ xP 1+η (f z 0 ,r ½ A 1 ) + inf ŷ∈B(x,r) δ ŷ P 1+η (f z 0 ,r ½ A 2 ) ≥ c x 0 η ∧ c y 0 η inf x∈B(x,r), ŷ∈B(y,r) µ 1+η x,ŷ (f z 0 ,r ) ≥ c x 0 η ∧ c y 0 η inf x∈B(x,r), ŷ∈B(y,r) µ 1+η x,ŷ (f z,(r+r 0 )/2 ),
where we used in the last line the inequality

f z 0 ,r (x) = (r -ρ(x, z 0 )) + ≥ (r -ρ(z, z 0 ) -ρ(x, z)) + ≥ f z 0 , r+r 0 2 (x).
Since (x, ŷ, z) ∈ E r 0 for all x ∈ B(x, r) and ŷ ∈ B(y, r), and since (x, y) → µ 1+η

x,y (f z 0 , r+r 0 2

) is continuous and positive on E r 0 (since r+r 0 2 > r 0 ), we deduce that µ t 0 x 0 ,y 0 (f z 0 ,r ) > 0 for all r > r 0 and t 0 ∈ (1, 2]. and hence (x 0 , y 0 , z 0 ) ∈ E r 0 . This concludes the proof of Lemma 4.2.

Recall the constant ε 2 introduced in (4.1). Fix s ≥ t 1 and t ∈ (1, 2]. We have, for all x, z ′ ∈ E,

δ x P t+s f ≥ M ½ z∈Mε 2 P t f (z) δ x P s (dz) ≥ M ½ z∈Mε 2 µ t z,z ′ (f ) δ x P s (dz).
Integrating both sides over z ′ ∈ M ε 2 with respect to δ y P s (dz ′ )/δ y P s ½ M , we deduce that

δ x P t+s f ≥ M ×M ½ z,z ′ ∈Mε 2 µ t z,z ′ (f ) δ x P s (dz) δ y P s (dz ′ ) δ y P s ½ M . But δ x P t+s ½ M ≤ δ x P s ½ M , hence δ x P t+s f δ x P t+s ½ M ≥ Mε 2 ×Mε 2 µ t z,z ′ (f ) δ x P s (dz) δ x P s ½ M δ y P s (dz ′ ) δ y P s ½ M .
Since t ∈ (1, 2], it follows from Lemma 4.2 that

m t = inf z,z ′ ∈Mε 2 µ t z,z ′ (M ) > 0,
and therefore, since s ≥ t 1 ,

m x,y t := Mε 2 ×Mε 2 µ t z,z ′ (M ) δ x P s (dz) δ x P s ½ M δ y P s (dz ′ ) δ y P s ½ M ≥ m t Mε 2 ×Mε 2 δ x P s (dz) δ x P s ½ M δ y P s (dz ′ ) δ y P s ½ M ≥ A 2 m t , (4.8)
by (4.1). Therefore, defining the probability measure

ν s,t x,y (•) = 1 m x,y t Mε 2 ×Mε 2 µ t z,z ′ (•) δ x P s (dz) δ x P s ½ M δ y P s (dz ′ ) δ y P s ½ M , (4.9)
we obtain that

δ x P t+s f δ x P t+s ½ M ≥ m x,y t ν s,t x,y (f ) ≥ A 2 m t ν s,t
x,y (f ).

Hence we have proved Condition (A1) for t 0 = t 1 + 2 (choosing s = t 1 and t = 2 and defining ν x,y = ν t 1 ,2 x,y ).

Balance condition on the absorption probabilities

Our goal is now to prove Condition (A2). We first show that the first step of our proof (where we show (4.1)) implies the following stronger version of the gradient estimates (3.3), but only for f = ½ M .

Proposition 4.3. There exists a constant c ′ > 0 such that, for all t > 0,

∇P t ½ M ∞ ≤ c ′ √ t ∧ 1 P t ½ M ∞ . (4.10)
Proof. This follows directly from the following computation: for all t ≥ t 0 = t 1 + 2,

P t ½ M (x) ≥ P x (X t-2 ∈ M ε 2 ) inf y∈Mε 2 P y (2 < τ ∂ ) ≥ AP x (t -2 < τ ∂ ) c ε 2 ,
by (4.1) and where c ε 2 := inf y∈Mε 2 P y (2 < τ ∂ ). Hence,

P t ½ M ∞ ≥ c ε 2 A P t-2 ½ M ∞ .
Hence it follows from (3.3) that, for all t > t 0 ,

∇P t ½ M ∞ = ∇P 2 (P t-2 ½ M ) ∞ ≤ c √ 2 P t-2 (½ M ) ∞ ≤ c √ 2c ε A P t ½ M ∞ .
Since in addition P t ½ M ∞ ≥ P t 0 ½ M ∞ for all t ≤ t 0 , we also deduce from (3.3) that

∇P t ½ M ∞ ≤ c √ t ∧ 1 ≤ c ( √ t ∧ 1) P t 0 ½ M ∞ P t ½ M ∞ .
This proposition implies that the function

x → P t ½ M (x) P t ½ M ∞ (4.11)
is c ′ -Lipschitz for all t ≥ 1. Since this functions vanishes on ∂M and its maximum over M is 1, we deduce that, for all t ≥ 1, the argmax of this function, denoted by z t , exists in M . Moreover, P t ½ M (x) P t ½ M ∞ ≥ f zt,r 0 (x) r 0 (4.12) for all x ∈ M , with r 0 := 1/c ′ . In particular, this implies that there exists ε ′ > 0 such that z t ∈ M ε ′ for all t ≥ 1. Without loss of generality, we assume that ε ′ ∈ (0, ε 2 ), where the constant ε 2 is taken from (4.1). Therefore, for all x, y ∈ M and for all t ≥ 1,

P ν t 1 ,2
x,y

(t < τ ∂ ) ≥ P t ½ M ∞ r 0 ν t 1 ,2
x,y (f zt,r 0 )

= P t ½ M ∞ r 0 m x,y 2 M ε ′ ×M ε ′ µ 2 z,z ′ (f zt,r 0 ) δ x P t 1 (dz) δ x P t 1 ½ M δ y P t 1 (dz ′ ) δ y P t 1 ½ M ,
where we used the definition (4.9) of the measure ν t 1 ,2 x,y . Now, it follows from the proof of Lemma 4.2 that (x, y, z) → µ 2

x,y (f z,r 0 ) is positive and continuous on M 3 . Hence, using that m x,y 2 ≤ 1 and defining c ′′ := inf (x,y,z)M 3 ε ′ µ 2 x,y (f z,r 0 ) > 0, we obtain

P ν t 1 ,2 x,y (t < τ ∂ ) ≥ c ′′ P t ½ M ∞ r 0
Mε 2 ×Mε 2 δ x P t 1 (dz)

δ x P t 1 ½ M δ y P t 1 (dz ′ ) δ y P t 1 ½ M ≥ c ′′ A 2 r 0 P t ½ M ∞ ,
where the last inequality follows from (4.1). This entails Condition (A2) for all t ≥ 1. For t ≤ 1, we make use of the following direct consequence of the proof Lemma 4.2:

c ′′′ := inf (x,y)∈M 2 ε µ 2
x,y (M ε ) > 0.

We deduce from (4.9) that ν t 1 ,2 x,y (M ε 2 ) ≥ c ′′′ /m x,y 2 and hence, for all t ≤ 1,

P ν t 1 ,2
x,y

(t < τ ∂ ) ≥ c ′′′ m x,y 2 inf z∈Mε 2 P z (t < τ ∂ ) ≥ c ′′′ m x,y 2 inf z∈Mε 2 P z (1 < τ ∂ ) ≥ c ′′′ m x,y 2 inf z∈Mε 2 P z (1 < τ ∂ ) sup z∈M P z (t < τ ∂ ) > 0.
This ends the proof of (A2) and hence of (1.1).

Contraction in total variation norm

Our aim is now to conclude the proof of Theorem 3.1 by proving (3.4). In order to complete this step, we make use of (2.6). Hence, we need to prove that for all probability measure µ on M , c(µ) = inf t≥0 P µ (t < τ ∂ ) P zt (t < τ ∂ ) ≥ cµ(ρ ∂M ) (4.13)

for some constant c > 0, where we recall that z t is defined as the argmax of (4.11). Enlarging t 0 and reducing c 1 and c 2 , one can assume without loss of generality that ν = ν x,y does not depend on x, y in (A) (this follows from the equivalence between (A), (A') and (A") proved in [START_REF] Champagnat | Exponential convergence to quasistationary distribution and Q-process[END_REF]Theorem 2.1]).

  2.3], a spectral gap result [4, Corollary 2.4] and the uniform exponential ergodicity of the Q-process [4, Theorem 3.1].

  S2. X is a diffusion process evolving in a bounded and connected domain M of R d with C 2 boundary ∂M and X satisfies the SDE dX t = s(X t )dB t + b(X t )dt, where (B t , t ≥ 0) is a r-dimensional standard Brownian motion, b : M → R d is bounded and continuous and s : M → R d×r is continuous, ss * is uniformly elliptic and for all r > 0, sup x,y∈M, |x-y|=r |s(x) -s(y)| 2 r ≤ g(r) (3.1)

Then, using (4.12) and (A1), we deduce that for all t ≥ t 0 + 1, P µ (t < τ ∂ ) = µ(P t 0 P t-t 0 ½ M )

Now, by Markov's property,

Since inf y∈M ε 2 (t 0 ) P y (t 0 < τ ∂ ) > 0, we deduce that there exists a constant c > 0 such that, for all x ∈ M ,

Integrating the last inequality with respect to µ, we deduce from (4.14) that, for all t ≥ t 0 + 1,

Since we have when t < t 0 + 1

the proof of (4.13) is completed. This ends the proof of Theorem 3.1.