Precise high moment asymptotics for parabolic Anderson model with log-correlated Gaussian field

TL;DR

This paper derives precise high moment asymptotics for the parabolic Anderson model driven by a log-correlated Gaussian field, revealing detailed behavior of moments in the large N limit.

Contribution

It provides the first rigorous asymptotic analysis of high moments for PAM with log-correlated Gaussian fields, connecting to free and Bessel fields.

Findings

Asymptotic formula for exponential moments involving Brownian motions and log-correlated fields.

Connection between the covariance structure and high moment behavior.

Enhanced understanding of the Feynman-Kac formula in this context.

Abstract

In this paper, we consider the continuous parabolic Anderson model (PAM) driven by a time-independent log-correlated Gaussian field (LGF). We obtain an asymptotic result of $$\mathbb{E}\exp\Bigg\{\frac{1}{2}\sum\limits_{ j,k=1}^N\int_0^t\int_0^t\gamma(B_j(s)-B_k(r))drds\Bigg\}\qquad(N\rightarrow \infty)$$ which is composed of the independent Brownian motions $\{B_j(s)\}$ and the function $\gamma$ approximating to a logarithmic potential at $0$, such as the covariances of massive free field and Bessel field. Based on the asymptotic result, we get the precise high moment asymptotics for Feynman-Kac formula of the PAM with LGF.