Feng-Yu Wang

Harnack and functional inequalities for generalized Mehler semigroups

Abstract Harnack inequalities are established for a class of generalized Mehler semigroups, which in particular imply upper bound estimates for the transition density. Moreover, Poincare and log-Sobolev inequalities are proved in terms of estimates for the square field operators. Furthermore, under a condition, well-known in the Gaussian case, we prove that generalized Mehler semigroups are strong Feller. The results are illustrated by concrete examples. In particular, we show that a generalized Mehler semigroup with an α-stable part is not hyperbounded but exponentially ergodic, and that the log-Sobolev constant obtained by our method in the special Gaussian case can be sharper than the one following from the usual curvature condition. Moreover, a Harnack inequality is established for the generalized Mehler semigroup associated with the Dirichlet heat semigroup on (0,1). We also prove that this semigroup is not hyperbounded.

Harnack inequality and applications for stochastic generalized porous media equations

By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the L P -norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity

Explicit conditions are presented for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.

FUNCTIONAL INEQUALITIES, SEMIGROUP PROPERTIES AND SPECTRUM ESTIMATES

This paper gives a reasonably self-contained account for some recent progress on functional inequalities, semigroup properties and spectrum estimates. Two sorts of functional inequalities are considered, they are actually equivalent and are general forms of Sobolev type inequalities. Semigroup properties, spectrum estimates and concentration of measures are described using these inequalities. Some criteria of functional inequalities and estimates of the spectral gap and the log-Sobolev constant are presented for diffusions on Riemannian manifolds and jump processes. Most yet unpublished results are reproved.

General formula for lower bound of the first eigenvalue on Riemannian manifolds

A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’ s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).

Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds

By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the study of heat kernel upper bound and contractivity properties of the semigroup. The main results are also extended to reflecting diffusion processes on Riemannian manifolds with nonconvex boundary.By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of \cite{W97} is also investigated.

Probability distance inequalities on Riemannian manifolds and path spaces

Abstract We construct Otto–Villanis coupling for general reversible diffusion processes on a Riemannian manifold. As an application, some new estimates are obtained for Wasserstein distances by using a Sobolev–Poincare type inequality introduced by Latala and Oleszkiewicz. The corresponding concentration estimates of the measure are presented. Finally, our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the L2-distance and the intrinsic distance. In particular, Talagrands inequality holds on the path space over a compact manifold.

Application of coupling methods to the Neumann eigenvalue problem

SummaryBy using coupling methods, some lower bounds are obtained for the first Neumann eigenvalue on Riemannian manifolds. This method is new and the results improve some known estimates. An example shows that our estimates can be sharp.

LOG-HARNACK INEQUALITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS IN HILBERT SPACES AND ITS CONSEQUENCES

A logarithmic type Harnack inequality is established for the semigroup of solutions to a stochastic differential equation in Hilbert spaces with non-additive noise. As applications, the strong Feller property as well as the entropy-cost inequality for the semigroup are derived with respect to the corresponding distance (cost function).

Harnack Inequality and Strong Feller Property for Stochastic Fast-Diffusion Equations ∗

Abstract As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333–1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev–Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.