Exponential contraction in Wasserstein distance on static and evolving manifolds

in Revue Roumaine de Mathématiques Pures et Appliquées (2021), 66(1), 107-129

In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not ... [more ▼]

In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not necessarily required to be non-negative. Compared to the results of Wang (2016), we focus on explicit estimates for the exponential contraction rate. Moreover, we show that our results extend to manifolds evolving under a geometric flow. As application, for the time-inhomogeneous semigroups, we obtain a gradient estimate with an exponential contraction rate under weak curvature conditions, as well as uniqueness of the corresponding evolution system of measures. [less ▲]

Functional inequalities on manifolds with non-convex boundary

in Science China Mathematics (2018), 61(8), 1421-1436

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non ... [more ▼]

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary. [less ▲]

Evolution systems of measures and semigroup properties on evolving manifolds

in Electronic Journal of Probability (2018), 23(20), 1-27

An evolving Riemannian manifold (M,g_t)_{t\in I} consists of a smooth d-dimensional manifold M, equipped with a geometric flow g_t of complete Riemannian metrics, parametrized by I=(-\infty,T). Given an ... [more ▼]

An evolving Riemannian manifold (M,g_t)_{t\in I} consists of a smooth d-dimensional manifold M, equipped with a geometric flow g_t of complete Riemannian metrics, parametrized by I=(-\infty,T). Given an additional C^{1,1} family of vector fields (Z_t)_{t\in I} on M. We study the family of operators L_t=\Delta_t +Z_t where \Delta_t denotes the Laplacian with respect to the metric g_t. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by L_t, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established. [less ▲]

Characterization of pinched Ricci curvature by functional inequalities

in Journal of Geometric Analysis (2018), 28(3), 2312-2345

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient ... [more ▼]

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, L^p-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities. [less ▲]

Diffusion semigroup on manifolds with time-dependent metrics

in Forum Mathematicum (2017), 29(4), 751-1002

Let $L_t:=\Delta_t +Z_t $, $t\in [0,T_c)$ on a differential manifold equipped with a complete geometric flow $(g_t)_{t\in [0,T_c)}$, where $\Delta_t$ is the Laplacian operator induced by the metric $g_t ... [more ▼]

Let $L_t:=\Delta_t +Z_t $, $t\in [0,T_c)$ on a differential manifold equipped with a complete geometric flow $(g_t)_{t\in [0,T_c)}$, where $\Delta_t$ is the Laplacian operator induced by the metric $g_t$ and $(Z_t)_{t\in [0,T_c)}$ is a family of $C^{1,\infty}$-vector fields. In this article, we present a number of equivalent inequalities for the lower bound curvature condition, which include gradient inequalities, transportation-cost inequalities, Harnack inequalities and other functional inequalities for the semigroup associated with diffusion processes generated by $L_t$. To this end, we establish the derivative formula for the associated semigroup and construct couplings for these diffusion processes by parallel displacement and reflection. [less ▲]

Algebraic Convergence Rate for Reflecting Diffusion Processes on Manifolds with Boundary

in Potential Analysis (2016), 44(1), 91-107

A criteria for the algebraic convergence rate of diffusion semigroups on manifolds with respect to some Lipschitz norms in L2-sense is presented by using a Lyapunov condition. As application, we apply it ... [more ▼]

A criteria for the algebraic convergence rate of diffusion semigroups on manifolds with respect to some Lipschitz norms in L2-sense is presented by using a Lyapunov condition. As application, we apply it to some diffusion processes with heavy tailed invariant distributions. This result is further extended to the reflecting diffusion processes on manifolds with non-convex boundary by using a conformal change of the metric. [less ▲]

An integration by parts formula on path space over manifolds carrying geometric flow

in Science China Mathematics (2015), 58(7), 1511--1522

Eigentime identity for one-dimensional diffusion processes

in Journal of Applied Probability (2015), 52(1), 224--237

L^2 Rate of Algebraic Convergence for Diffusion Processes on Non-Convex Manifold

in Chinese Journal of Applied Probability and Statistics (2015), 31(5), 495-502

Algebraic convergence in L2-sense is studied for the reflecting diffusion processes on noncompact manifold with non-convex boundary. A series of su cient and necessary conditions for the algebraic ... [more ▼]

Algebraic convergence in L2-sense is studied for the reflecting diffusion processes on noncompact manifold with non-convex boundary. A series of su cient and necessary conditions for the algebraic convergence are presented. [less ▲]

$L^2$-algebraic decay rate for transient birth-death processes

in Chinese Annals of Mathematics (2012), 33(4), 583--594