References of "Cheng, Li Juan 50009204"
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See detailUniform gradient estimates on manifolds with a boundary and applications
Cheng, Li Juan UL; Thalmaier, Anton UL; Thompson, James UL

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 ▲]

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See detailFunctional inequalities on manifolds with non-convex boundary
Cheng, Li Juan UL; Thalmaier, Anton UL; Thompson, James UL

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 ▲]

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See detailQuantitative C1-estimates by Bismut formulae
Cheng, Li Juan UL; Thalmaier, Anton UL; Thompson, James UL

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 ▲]

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See detailEvolution systems of measures and semigroup properties on evolving manifolds
Cheng, Li Juan UL; Thalmaier, Anton UL

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 ▲]

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See detailSpectral gap on Riemannian path space over static and evolving manifolds
Cheng, Li Juan UL; Thalmaier, Anton UL

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 ▲]

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See detailCharacterization of pinched Ricci curvature by functional inequalities
Cheng, Li Juan UL; Thalmaier, Anton UL

in Journal of Geometric Analysis (The) (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 ▲]

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See detailReflecting diffusion semigroup on manifolds carrying geometric flow
Cheng, Li Juan UL; Zhang, Kun

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 ▲]

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See detailDiffusion semigroup on manifolds with time-dependent metrics
Cheng, Li Juan UL

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 ▲]

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See detailTransportation-cost inequalities on path spaces over manifolds carrying geometric flows
Cheng, Li Juan UL

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 ▲]

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See detailAlgebraic Convergence Rate for Reflecting Diffusion Processes on Manifolds with Boundary
Cheng, Li Juan UL; Wang, Yingzhe

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 ▲]

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See detailWeak poincaré inequality for convolution probability measures
Cheng, Li Juan UL; Zhang, Shao Qin

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.

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See detailThe radial part of Brownian motion with respect to $\CalL$-distance under Ricci flow
Cheng, Li Juan UL

in J. Theoret. Probab. (2015), 28(2), 449--466

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See detailAn integration by parts formula on path space over manifolds carrying geometric flow
Cheng, Li Juan UL

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

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See detailA probabilistic method for gradient estimates of some geometric flows
Chen, Xin; Cheng, Li Juan UL; Mao, Jing

in Stochastic Process. Appl. (2015), 125(6), 2295--2315

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See detailL^2 Rate of Algebraic Convergence for Diffusion Processes on Non-Convex Manifold
Cheng, Li Juan UL; Wang, Yingzhe

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 ▲]

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See detailEigentime identity for one-dimensional diffusion processes
Cheng, Li Juan UL; Mao, Yong-Hua

in J. Appl. Probab. (2015), 52(1), 224--237

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See detailA probabilistic approach for gradient estimates on time-inhomogeneous manifolds
Cheng, Li Juan UL

in Statist. Probab. Lett. (2014), 88

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See detail$L^2$-algebraic decay rate for transient birth-death processes
Cheng, Li Juan UL; Wang, Yingzhe

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

Detailed reference viewed: 83 (9 UL)