Brownian motion in attenuated or renormalized inverse-square Poisson potential

TL;DR

This paper studies the behavior of Brownian motion in a random potential with inverse-square singularities, establishing existence and asymptotics of solutions, and resolving a critical case in three dimensions.

Contribution

It introduces a framework for analyzing Brownian motion in inverse-square Poisson potentials, including renormalization in 3D, and determines large-time asymptotics for solutions.

Findings

Proved existence of solutions for the parabolic Anderson problem with inverse-square potentials.

Derived large-time asymptotics of solutions using Feynman-Kac representation.

Resolved the critical parameter case in three dimensions for the renormalized potential.

Abstract

We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^d$, $d \geq 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x) \approx \theta |x|^{-2}$ near the origin, where $\theta \in (0,(d-2)^2/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in $d=3$ the problem with critical parameter $\theta = 1/16$, left open by Chen and Rosinski in [arXiv:1103.5717].