Path-by-path regularization by noise for scalar conservation laws

TL;DR

This paper demonstrates how noise can regularize scalar conservation laws on a path-by-path basis, extending previous results to fractional Brownian motion and introducing a new scaling property that ensures regularization.

Contribution

It introduces a novel path-by-path regularization framework for scalar conservation laws driven by noise, including fractional Brownian motion, and proposes a new scaling property that guarantees regularization effects.

Findings

Regularization by noise for scalar conservation laws proven on a path-by-path basis.

Extension of regularization results to fractional Brownian motion.

Introduction of a new path-by-path scaling property ensuring regularization.

Abstract

We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [Gess, Souganidis; Comm. Pure Appl. Math. (2017)]. In addition, we introduce a new path-by-path scaling property which is shown to be sufficient to imply regularizing effects.