Continuous dependence estimate for conservation laws with L\'{e}vy noise

TL;DR

This paper derives explicit continuous dependence estimates for multidimensional stochastic conservation laws driven by Lévy noise, using BV estimates and analyzing the effects of noise dependence on solutions.

Contribution

It provides new continuous dependence and error estimates for entropy solutions of stochastic conservation laws with Lévy noise, including fractional BV estimates.

Findings

Explicit continuous dependence estimates derived.

Error estimates for stochastic vanishing viscosity method established.

Fractional BV estimates obtained when noise depends on solution and space.

Abstract

We are concerned with multidimensional stochastic balance laws driven by L\'{e}vy processes. 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\'{e}vy noise only depends on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional $BV$ estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable.