Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise
This provides a theoretical estimate for a class of stochastic PDEs, but the result is incremental for specialists in stochastic conservation laws.
The authors derive an explicit continuous dependence estimate on the nonlinearities of entropy solutions for a degenerate parabolic-hyperbolic equation driven by Levy noise, and use it to show an error estimate for the stochastic vanishing viscosity method. They also establish a fractional BV estimate when noise coefficients depend on both solution and spatial variable.
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.