Conservation laws driven by Lévy white noise

Abstract

We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itô-Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the $L^1$-contraction principle. Finally, the $L^1$ contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.

46 pages