Functional Inequalities for Particle Systems on Polish Spaces

Abstract

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.