Stochastic conservation laws with Poisson noise: Well-posedness of càdlàg entropy solutions and stability of sample paths

Abstract

Our focus here is stochastic conservation laws driven by pure-jump type noise. We wish to set the stochastic entropy solution framework for such problems on a stronger footing. This is done by emphasising on the regularity of sample paths of a prospective stochastic entropy solution. We first prove the well-posedness of stochastic entropy solutions that are càdlàg and adapted stochastic processes with values in appropriate function spaces. This inherent càdlàg property then enables us to derive stability results for sample paths in terms of Skorohod-type metric, the natural metric in the path space. We achieve this by establishing refined path-based maximal-type stability estimates for the viscous approximation. Moreover, the rate of convergence for the sample paths of the viscous perturbation is computed explicitly. In addition, we are able to get rid of some crucial technical requirements and claim well-posedness for a wider class of problems.