Regularization by random translation of potentials for the continuous PAM and related models in arbitrary dimension

TL;DR

This paper introduces a noise-based regularization technique for the continuous parabolic Anderson model using fractional Brownian motion shifts, enabling well-posedness in any dimension without renormalization.

Contribution

It demonstrates that small Hurst parameter shifts of potentials ensure well-posedness and stability for the PAM in arbitrary dimensions, avoiding renormalization.

Findings

Establishes well-posedness for PAM with fractional Brownian motion shifts

Provides a Feynman-Kac formula for the regularized solution

Shows stability of solutions under potential shifts

Abstract

We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.