Gibbs measures with double stochastic integrals on a path space

TL;DR

This paper studies Gibbs measures associated with Brownian motion where the interaction energy involves a double stochastic integral, proving the existence of an infinite volume limit in certain quantum electrodynamics models.

Contribution

It introduces a new class of Gibbs measures with double stochastic integrals and establishes their infinite volume limit in the context of quantum electrodynamics.

Findings

Existence of infinite volume Gibbs measures for models with double stochastic integrals.

Application to the Pauli-Fierz model in nonrelativistic quantum electrodynamics.

Extension of Gibbs measure theory to complex stochastic interactions.

Abstract

We investigate Gibbs measures relative to Brownian motion in the case when the interaction energy is given by a double stochastic integral. In the case when the double stochastic integral is originating from the Pauli-Fierz model in nonrelativistic quantum electrodynamics, we prove the existence of its infinite volume limit.