Gibbs measures on Brownian paths: Theory and applications

TL;DR

This paper reviews the theory of Gibbs measures on Brownian paths, discussing their existence, uniqueness, and properties, and presents a functional central limit theorem for increment-dependent energies.

Contribution

It provides a comprehensive review of Gibbs measures on Brownian motion, including new results on existence, uniqueness, and a functional central limit theorem.

Findings

Existence and path properties of Gibbs measures established.

Conditions for uniqueness and non-uniqueness analyzed.

A functional central limit theorem for increment-dependent energies proved.

Abstract

We review our investigations on Gibbs measures relative to Brownian motion, in particular the existence of such measures and their path properties, uniqueness, resp. non-uniqueness. For the case when the energy only depends on increments, we present a functional central limit theorem. We also explain connections with other work and state open problems of interest.