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In order to describe L 2 -convergence rates slower than exponential, the weak Poincare inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare inequality can be determined by each other. Conditions for the weak Poincare inequality to hold are presented, which are easy to check and which hold in many applications. The weak Poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes. Some typical examples are given to illustrate the general results. In particular, our results are applied to the stochastic quantization of field theory in finite volume. Moreover, a sharp criterion of weak Poincare inequalities is presented for Poisson measures on configuration spaces. Academic Press

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ -finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N -functions in the theory of Orlicz spaces.

This paper presents a dimension-free Harnack type inequality for heat semigroups on manifolds, from which a dimension-free lower bound is obtained for the logarithmic Sobolev constant on compact manifolds and a new criterion is proved for the logarithmic Sobolev inequalities (abbrev. LSI) on noncompact manifolds. As a result, it is shown that LSI may hold even though the curvature of the operator is negative everywhere.

In terms of the equivalence of Poincare inequality and the existence of spectral gap, the super-Poincare inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for super-Poincare inequalities is studied, the resulting estimates extend previous ones for Poincare and log-Sobolev inequalities. Academic Press

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup P * t on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup P * t using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.

Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimensionfree Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by −c(1 + ρ 2 o), where c > 0 is a constant and ρ o is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li-Yau type heat kernel bound is presented for such semigroups.

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