Rough linear PDE's with discontinuous coefficients - existence of solutions via regularization by fractional Brownian motion

TL;DR

This paper demonstrates that adding fractional Brownian motion noise can regularize ill-posed linear PDEs with discontinuous coefficients, ensuring the existence of solutions for almost all noise paths.

Contribution

It introduces a novel regularization approach for linear PDEs with discontinuous drifts using fractional Brownian motion, establishing existence results.

Findings

Existence of solutions for PDEs with discontinuous coefficients under fractional Brownian motion perturbation.

Fractional Brownian motion regularizes ill-posed PDEs.

Solutions exist for almost all noise paths.

Abstract

We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.