Laplace operators in gamma analysis

TL;DR

This paper investigates Laplace-type operators associated with gamma measures on the cone of discrete Radon measures, proving their essential self-adjointness on a set of test functions, which advances the understanding of analysis on measure spaces.

Contribution

It introduces and studies a class of Laplace operators in the space of measures with gamma distribution, establishing their essential self-adjointness, a key property for analysis and stochastic processes.

Findings

Operators are well-defined on test functions

Proved essential self-adjointness of these operators

Provides foundation for further analysis on measure spaces

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 L\'evy 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 $\mathbb K(\mathbb R^d)$.