Functional Inequalities for Weighted Gamma Distribution on the Space of Finite Measures

Abstract

Let $\MM$ be the space of finite measures on a Locally compact Polish space, and let $\BG$ be the Gamma distribution on $\MM$ with intensity measure $\nu\in \MM$. Let $\nn^{ext}$ be the extrinsic derivative with tangent bundle $T\MM= \cup_{\eta\in\MM} L^2(\eta)$, and let $\GA: T\MM\to T\MM$ be measurable such that $\GA_\eta$ is a positive definite linear operator on $L^2(\eta)$ for every $\eta\in \MM$. Moreover, for a measurable function $V$ on $\MM$, let $\d\BG^V= \e^V\d\BG$. We investigate the Poincar\'e, weak Poincar\'e and super Poincar\'e inequalities for the Dirichlet form $$\EE_{\GA,V}(F,G):= \int_\MM \<\GA_\eta\nn^{ext}F(\eta), \nn^{ext}G(\eta)\>_{L^2(\eta)}\, \d\BG^V(\eta),$$ which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.