A Meta Theorem for nonlinear stochastic coupled systems: Application to stochastic chemotaxis-Stokes porous media

Abstract

The purpose of the paper is twofold. Firstly, we want to present a Meta Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic differential equations. The idea is first to split the system by rewriting the non-linear part in a linear part acting on a given process $\xi$. This is done in such a way that the fixpoint with respect to $\xi$ would be the solution. However, to show the well posedness of the {\sl linearized} system, one needs a cut-off argument. Under which conditions one can handle the limits of the cut-off parameter to get in the end a martingale solution of the original system is given in the Meta-Theorem. Secondly, we want to verify the full applicability of the Meta Theorem by showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamic.