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

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.

A pathwise regularization by noise phenomenon for the evolutionary p-Laplace equation

“…An extension of pathwise regularization techniques to the two parameter setting with applications to regularization by noise for the stochastic wave equation was established in [4]. Pathwise regularization by noise for the multiplicative stochastic heat equations with spatial white noise was treated in [2,9].…”

“…establishing the first claim. Next, we take a look at (2). By part (1) the right-hand side of (2.5) is weakly differentiable in L 2…”