Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise

TL;DR

This paper derives continuous dependence estimates for solutions of a multidimensional degenerate parabolic-hyperbolic equation driven by Levy noise, providing error bounds for stochastic viscosity methods and fractional BV estimates.

Contribution

It introduces explicit continuous dependence estimates for entropy solutions with Levy noise depending only on the solution, and extends to fractional BV estimates when noise depends on space and solution.

Findings

Explicit continuous dependence estimate derived

Error estimate for stochastic vanishing viscosity method established

Fractional BV estimate obtained for more general noise dependence

Abstract

In this article, we are concerned with a multidimensional degenerate parabolic-hyperbolic equation driven by Levy 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 Levy noise depends only 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 coefficients depend on both the solution and spatial variable.