Rough path continuity equations with discontinuous coefficients - regularization by fractional Brownian motion

TL;DR

This paper studies stochastic continuity equations with discontinuous drifts, showing that fractional Brownian motion regularizes the problem and ensures solutions exist for almost all paths.

Contribution

It demonstrates that fractional Brownian motion can regularize stochastic continuity equations with discontinuous coefficients, guaranteeing solutions for almost all paths.

Findings

Existence of solutions for almost all fractional Brownian motion paths.

Regularization effect of fractional Brownian motion on discontinuous coefficients.

Extension of stochastic continuity equations to irregular drifts.

Abstract

We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.