Gibbs measures on mutually interacting Brownian paths under singularities

TL;DR

This paper analyzes Gibbs measures on two mutually interacting Brownian paths with singular potentials, establishing large deviation principles and asymptotic behaviors, and applies results to the parabolic Anderson model to understand intermittency.

Contribution

It introduces a novel compactification for analyzing mutually interacting Brownian paths with singular potentials and proves a strong large deviation principle in this setting.

Findings

Established a strong large deviation principle for mutual Brownian path interactions.

Derived asymptotic path behaviors under Gibbs measures with singular interactions.

Computed annealed Lyapunov exponents for the parabolic Anderson model, confirming intermittency.

Abstract

We are interested in the analysis of Gibbs measures defined on two independent Brownian paths in $\mathbb R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $\int\int_{\mathbb R^{2d}} V(x-y) \mu(d x)\nu(d y)$ with two probability measures $\mu$ and $\nu$ representing the occupation measures of two independent Brownian motions. We will be interested in class of potentials $V$ which are {\it{singular}}, e.g., Dirac or Coulomb type interactions in $\mathbb R^3$, or the correlation function of the parabolic Anderson problem with white noise potential. The mutual interaction of the Brownian paths inspires a compactification of the quotient space of orbits of product measures, which is structurally different from the self-interacting case introduced in \cite{MV14}, owing to the lack of shift-invariant structure in the mutual interaction. We prove a {\it{strong large deviation principle}} for the product measures of two Brownian occupation measures in such a compactification, and derive asymptotic path behavior under Gibbs measures on Wiener paths arising from mutually attracting singular interactions. For the spatially smoothened parabolic Anderson model with white noise potential, our analysis allows a direct computation of the annealed Lyapunov exponents and a strict ordering of them implies the {\it{intermittency effect}} present in the smoothened model.