On stochastic conservation laws and Malliavin calculus

TL;DR

This paper extends the theory of stochastic conservation laws by integrating Malliavin calculus into the entropy condition framework, establishing existence and uniqueness results for solutions with Malliavin differentiable random constants.

Contribution

It introduces a novel generalization of Kruzhkov entropy conditions using Malliavin differentiable constants, advancing the mathematical understanding of stochastic conservation laws.

Findings

Established existence and uniqueness of solutions.

Provided a new perspective on stochastic entropy conditions.

Linked Malliavin calculus with stochastic conservation law theory.

Abstract

For stochastic conservation laws driven by a semilinear noise term, we propose a generalization of the Kru\v{z}kov entropy condition by allowing the Kru\v{z}kov constants to be Malliavin differentiable random variables. Existence and uniqueness results are provided. Our approach sheds some new light on the stochastic entropy conditions put forth by Feng and Nualart [J. Funct. Anal., 2008] and Bauzet, Vallet, and Wittbold [J. Hyperbolic Differ. Equ., 2012].