Scalar conservation laws with rough flux and stochastic forcing

TL;DR

This paper develops a framework for scalar conservation laws influenced by rough paths and stochastic forcing, establishing well-posedness and continuous dependence using advanced analytical tools.

Contribution

It introduces a kinetic formulation for conservation laws with rough flux and stochastic forcing, proving existence, uniqueness, and stability of solutions.

Findings

Established well-posedness of solutions

Proved $L^1$-contraction property

Extended analysis to flux driven by Brownian motion with Lévy area

Abstract

In this paper, we study scalar conservation laws where the flux is driven by a geometric H\"older $p$-rough path for some $p\in (2,3)$ and the forcing is given by an It\^o stochastic integral driven by a Brownian motion. In particular, we derive the corresponding kinetic formulation and define an appropriate notion of kinetic solution. In this context, we are able to establish well-posedness, i.e. existence, uniqueness and the $L^1$-contraction property that leads to continuous dependence on initial condition. Our approach combines tools from rough path analysis, stochastic analysis and theory of kinetic solutions for conservation laws. As an application, this allows to cover the case of flux driven for instance by another (independent) Brownian motion enhanced with L\'evy's stochastic area.