Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation

TL;DR

This paper demonstrates that irregular perturbations can ensure well-posedness of the stochastic heat equation with multiplicative noise, even with distributional coefficients, by establishing pathwise regularization effects.

Contribution

It introduces a novel approach of using irregular continuous paths to regularize the stochastic heat equation with multiplicative noise, extending well-posedness to distributional coefficients.

Findings

Irregular perturbations establish well-posedness of the stochastic heat equation.

Regularity of the averaged field for Lévy fractional stable motion is proven.

Perturbation by irregular paths can regularize equations with generalized function coefficients.

Abstract

Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a L\'evy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.