# The Cauchy problem for a fractional conservation laws driven by L\'evy noise

## TL;DR

This paper investigates the mathematical properties of multidimensional fractional conservation laws influenced by Lévy noise, establishing existence, uniqueness, and error estimates for solutions using entropy methods and BV estimates.

## Contribution

It introduces a new framework for analyzing stochastic fractional conservation laws driven by Lévy processes, including explicit dependence and regularization results.

## Key findings

- Existence and uniqueness of solutions established
- Derived continuous dependence and error estimates
- Analyzed vanishing non-local regularization

## Abstract

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by L\'evy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that L\'evy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws.