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Functional inequalities on path space of sub-Riemannian manifolds and applications Cheng, Li Juan ; ; Thalmaier, Anton in Nonlinear Analysis: Theory, Methods and Applications (2021), 210(112387), 1-30 We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians ... [more ▼] We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in L^2. With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein-Uhlenbeck operator. [less ▲] Detailed reference viewed: 175 (37 UL)Exponential contraction in Wasserstein distance on static and evolving manifolds Cheng, Li Juan ; Thalmaier, Anton ; 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 ▲] Detailed reference viewed: 181 (35 UL)Uniform gradient estimates on manifolds with a boundary and applications Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Analysis and Mathematical Physics (2018), 8(4), 571-588 We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using ... [more ▼] We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux's argument, and uniform quantitative gradient estimates, firstly for bounded C^2 functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians. [less ▲] Detailed reference viewed: 265 (67 UL)Functional inequalities on manifolds with non-convex boundary Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James 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 ▲] Detailed reference viewed: 294 (46 UL)Quantitative C1-estimates by Bismut formulae Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Journal of Mathematical Analysis and Applications (2018), 465(2), 803-813 For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a ... [more ▼] For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a condition for the zero-mean value property of Δu. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings. [less ▲] Detailed reference viewed: 367 (65 UL)Evolution systems of measures and semigroup properties on evolving manifolds Cheng, Li Juan ; Thalmaier, Anton 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 ▲] Detailed reference viewed: 361 (69 UL)Spectral gap on Riemannian path space over static and evolving manifolds Cheng, Li Juan ; Thalmaier, Anton in Journal of Functional Analysis (2018), 274(4), 959-984 In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along ... [more ▼] In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber’s recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space. [less ▲] Detailed reference viewed: 343 (43 UL)Characterization of pinched Ricci curvature by functional inequalities Cheng, Li Juan ; Thalmaier, Anton 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 ▲] Detailed reference viewed: 384 (54 UL)Reflecting diffusion semigroup on manifolds carrying geometric flow Cheng, Li Juan ; in Journal of Theoretical Probability (2017), 30(4), 1334-1368 Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differentiable manifold $M$ with boundary $\partial M$, where $\Delta_t$ is the Laplacian operator, induced by a time dependent metric $g_t ... [more ▼] Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differentiable manifold $M$ with boundary $\partial M$, where $\Delta_t$ is the Laplacian operator, induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We first establish the derivative formula for the associated reflecting diffusion semigroup generated by $L_t$; then construct the couplings for the reflecting $L_t$-diffusion processes by parallel displacement and reflection, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup; and finally, by using the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary, including the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups. [less ▲] Detailed reference viewed: 171 (32 UL)Diffusion semigroup on manifolds with time-dependent metrics Cheng, Li Juan 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 ▲] Detailed reference viewed: 275 (66 UL)Transportation-cost inequalities on path spaces over manifolds carrying geometric flows Cheng, Li Juan in Bulletin des Sciences Mathématiques (2016), 140(5), 541-561 Let Lt:=Δt+ZtLt:=Δt+Zt for a C1,1C1,1-vector field Z on a differential manifold M possibly with a boundary ∂M , where ΔtΔt is the Laplacian operator induced by a time dependent metric gtgt differentiable ... [more ▼] Let Lt:=Δt+ZtLt:=Δt+Zt for a C1,1C1,1-vector field Z on a differential manifold M possibly with a boundary ∂M , where ΔtΔt is the Laplacian operator induced by a time dependent metric gtgt differentiable in t∈[0,Tc)t∈[0,Tc). In this article, by constructing suitable coupling, transportation-cost inequalities on the path space of the (reflecting if ∂M≠∅∂M≠∅) diffusion process generated by LtLt are proved to be equivalent to a new curvature lower bound condition and the convexity of the geometric flow (i.e., the boundary keeps convex). Some of them are further extended to non-convex flows by using conformal changes of the flows. As an application, these results are applied to the Ricci flow with the umbilic boundary. [less ▲] Detailed reference viewed: 162 (27 UL)Algebraic Convergence Rate for Reflecting Diffusion Processes on Manifolds with Boundary Cheng, Li Juan ; 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 ▲] Detailed reference viewed: 137 (8 UL)Weak poincaré inequality for convolution probability measures Cheng, Li Juan ; E-print/Working paper (2016) In this article, weak Poincaré inequalities are established for convolution measures by using Lyapunov conditions. As applications, these results are further applied to some explicit cases. Detailed reference viewed: 90 (12 UL)An integration by parts formula on path space over manifolds carrying geometric flow Cheng, Li Juan in Science China Mathematics (2015), 58(7), 1511--1522 Detailed reference viewed: 101 (8 UL)A probabilistic method for gradient estimates of some geometric flows ; Cheng, Li Juan ; in Stochastic Processes and Their Applications (2015), 125(6), 2295--2315 Detailed reference viewed: 201 (29 UL)Eigentime identity for one-dimensional diffusion processes Cheng, Li Juan ; in Journal of Applied Probability (2015), 52(1), 224--237 Detailed reference viewed: 186 (15 UL)The radial part of Brownian motion with respect to $\CalL$-distance under Ricci flow Cheng, Li Juan in Journal of Theoretical Probability (2015), 28(2), 449--466 Detailed reference viewed: 130 (11 UL)L^2 Rate of Algebraic Convergence for Diffusion Processes on Non-Convex Manifold Cheng, Li Juan ; 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 ▲] Detailed reference viewed: 62 (7 UL)A probabilistic approach for gradient estimates on time-inhomogeneous manifolds Cheng, Li Juan in Statistics and Probability Letters (2014), 88 Detailed reference viewed: 200 (26 UL)$L^2$-algebraic decay rate for transient birth-death processes Cheng, Li Juan ; in Chinese Annals of Mathematics (2012), 33(4), 583--594 Detailed reference viewed: 168 (9 UL) |
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