Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Let P t be the (Neumann) diffusion semigroup P t generated by a weighted Laplacian on a complete connected Riemannian manifold M without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant ≪> 0 if and only if W p ( μ 1 P t , μ 2 P t ) ≤ e − ≪ t W p ( μ 1 , μ 2 ) , t ≥ 0 , p ≥ 1 $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$ holds for all probability measures μ 1 and μ 2 on M , where W p is the L p Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction W p ( μ 1 P t , μ 2 P t ) ≤ c e − ≪ t W p ( μ 1 , μ 2 ) , p ≥ 1 , t ≥ 0 $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq ce^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ p \geq 1, t\geq 0$$ for some constants c ,≪> 0 for a class of diffusion semigroups with negative curvature where the constant c is essentially larger than 1. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise.

[1]  M. Émery,et al.  Hypercontractivité de semi-groupes de diffusion , 1984 .

[2]  Barry Simon,et al.  Ultracontractivity and the Heat Kernel for Schrijdinger Operators and Dirichlet Laplacians , 1987 .

[3]  Wilfrid S. Kendall,et al.  Nonnegative ricci curvature and the brownian coupling property , 1986 .

[4]  M. Émery.  Stochastic Calculus in Manifolds , 1989 .

[5]  L. Gross.  Logarithmic Sobolev inequalities and contractivity properties of semigroups , 1993 .

[6]  Daniel W. Stroock,et al.  Moment estimates derived from Poincar'e and log-arithmic Sobolev inequalities , 1994 .

[7]  Feng-Yu Wang.  Application of coupling methods to the Neumann eigenvalue problem , 1994 .

[8]  S. Aida,et al.  Logarithmic Sobolev Inequalities and Exponential Integrability , 1994 .

[9]  Feng-Yu Wang,et al.  Logarithmic Sobolev inequalities on noncompact Riemannian manifolds , 1997 .

[10]  Feng-Yu Wang,et al.  Estimates of Logarithmic Sobolev Constant: An Improvement of Bakry–Emery Criterion , 1997 .

[11]  Feng-Yu Wang,et al.  Estimation of spectral gap for elliptic operators , 1997 .

[12]  W. Stannat.  (Nonsymmetric) Dirichlet operators on $L^1$ : existence, uniqueness and associated Markov processes , 1999 .

[13]  Feng-Yu Wang,et al.  Functional Inequalities for Empty Essential Spectrum , 2000 .

[14]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[15]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[16]  S. Bobkov,et al.  Hypercontractivity of Hamilton-Jacobi equations , 2001 .

[17]  M. Röckner,et al.  Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups , 2001 .

[18]  C. Villani.  Topics in Optimal Transportation , 2003 .

[19]  M. Röckner,et al.  Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds , 2003 .

[20]  M. Röckner,et al.  On the Spectrum of a Class of Non‐Sectorial Diffusion Operators , 2004 .

[21]  Feng-Yu Wang.  Probability distance inequalities on Riemannian manifolds and path spaces , 2004 .

[22]  Karl-Theodor Sturm,et al.  Transport inequalities, gradient estimates, entropy and Ricci curvature , 2005 .

[23]  王 风雨.  Functional inequalities, Markov semigroups and spectral theory , 2005 .

[24]  Hamilton-Jacobi semi-groups in infinite dimensional spaces , 2006 .

[25]  Anton Thalmaier,et al.  Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below , 2006 .

[26]  Feng-Yu Wang,et al.  Gradient estimates for diffusion semigroups with singular coefficients , 2006 .

[27]  Feng-Yu Wang.  Log-Sobolev inequalities: Different roles of Ric and Hess , 2007, 0712.3143.

[28]  Feng-Yu Wang,et al.  Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds , 2009, 0911.1644.

[29]  Kazumasa Kuwada,et al.  Duality on gradient estimates and Wasserstein controls , 2009, 0910.1741.

[30]  Feng-Yu Wang.  Harnack inequalities on manifolds with boundary and applications , 2009, 0908.2888.

[31]  Feng-Yu Wang,et al.  Log-Harnack Inequality for Stochastic Differential Equations in Hilbert Spaces and its Consequences , 2009, 0911.0290.

[32]  Chenggui Yuan,et al.  Harnack inequalities for functional SDEs with multiplicative noise and applications , 2010, 1012.5688.

[33]  A. Eberle.  Reflection coupling and Wasserstein contractivity without convexity , 2011 .

[34]  M. Ledoux,et al.  On Harnack inequalities and optimal transportation , 2012, 1210.4650.

[35]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[36]  Feng-Yu Wang.  Criteria of spectral gap for Markov operators , 2013, 1305.4460.

[37]  Kazumasa Kuwada.  Gradient estimate for Markov kernels, Wasserstein control and Hopf-Lax formula (Potential Theory and its Related Fields) , 2013 .

[38]  Feng-Yu Wang.  Analysis for Diffusion Processes on Riemannian Manifolds , 2013 .

[39]  Jian Wang,et al.  L-Wasserstein Distance for Diffusion Processes , 2014 .

[40]  Feng-Yu Wang,et al.  Asymptotic couplings by reflection and applications for nonlinear monotone SPDES , 2014, 1407.3522.

[41]  Laurent Miclo,et al.  On hyperboundedness and spectrum of Markov operators , 2014, Inventiones mathematicae.

[42]  A. Eberle.  Couplings, distances and contractivity for diffusion processes revisited , 2013 .

[43]  T. Zhukovskaya,et al.  Functional Inequalities , 2021, Inequalities in Analysis and Probability.