Towards on convolutions on configuration spaces. II. Spaces of locally finite configurations

TL;DR

This paper explores convolutions of probability measures on spaces of locally finite configurations, examining their relation to correlation measures, Gibbs measures, and invariant measures under certain operators.

Contribution

It introduces a detailed analysis of convolutions on configuration spaces and their connections to correlation measures and Gibbs measures, extending prior theoretical frameworks.

Findings

Convolution of Gibbs measures is characterized.

Connections between invariant measures and correlation functions are established.

Theoretical insights into operators acting on correlation measures are provided.

Abstract

In the second part of the paper we consider a convolution of probability measures on spaces of locally finite configurations (subsets of a phase space) as well as their connection with the convolution of the corresponding correlation measures and functionals. In particular, the convolution of Gibbs measures is studied. We describe also a connection between invariant measures with respect to some operator and properties of the corresponding image of this operator on correlation functions.