Laplace operators in gamma analysis

Abstract

Let $\mathbb{K}({\mathbb{R}}^d)$ denote the cone of discrete Radon measures on ${\mathbb{R}}^d$. The gamma measure $\mathcal{G}$ is the probability measure on $\mathbb{K}({\mathbb{R}}^d)$ which is a measure-valued Lévy process with intensity measure $s^{-1}{e}^{-s}ds$ on $(0,\infty )$. We study a class of Laplace-type operators in $L^2(\mathbb{K}({\mathbb{R}}^d),\mathcal{G})$. These operators are defined as generators of certain (local) Dirichlet forms. The main result of the papers is the essential self-adjointness of these operators on a set of `test' cylinder functions on $\mathbb{K}({\mathbb{R}}^d)$.