Functional Inequalities for Particle Systems on Polish Spaces

Various Poincare-Sobolev type inequalities are studied for a reaction-diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction-diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces $E^n (n \geq 1)$ which determine the motion of particles, and the reaction part induced by a $Q$-process on $\mathbb Z_+$ and a sequence of reference probability measures, where the $Q$-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincare and weak Poincare inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding $Q$-process. But under a mild condition, stronger inequalities rely on both parts: the reaction-diffusion Dirichlet form satisfies a super Poincare inequality (e.g. the log-Sobolev inequality) if and only if so do both the corresponding $Q$-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results.Comment: 22 pages, BiBoS-Preprint no. 04-08-153, to appear in Potential Analysi

General Extinction Results for Stochastic Partial Differential Equations and Applications

Let $L$ be a positive definite self-adjoint operator on the $L^2$-space associated to a \si-finite measure space. Let $H$ be the dual space of the domain of $L^{1/2}$ w.r.t. $L^2(\mu)$. By using an It\^o type inequality for the $H$-norm and an integrability condition for the hyperbound of the semigroup P_t:=\e^{-Lt}, general extinction results are derived for a class of continuous adapted processes on $H$. Main applications include stochastic and deterministic fast diffusion equations with fractional Laplacians. Furthermore, we prove exponential integrability of the extinction time for all space dimensions in the singular diffusion version of the well-known Zhang-model for self-organized criticality, provided the noise is small enough. Thus we obtain that the system goes to the critical state in finite time in the deterministic and with probability one in finite time in the stochastic case.Comment: 19 page

Stochastic Generalized Porous Media and Fast Diffusion Equations

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 $\sigma$-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.Comment: 36 pages, BiBoS-Preprint No. 06-02-20

Large Deviations for Stochastic Generalized Porous Media Equations

The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.Comment: 15 pages; BiBoS-Preprint No. 05-11-196; publication in preparatio

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)

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.Comment: 15 pages; BiBoS-Preprint No. 04-09-157; to appear in Commun. PD