Multicomponent systems like gas mixtures, biological cells, and population dynamics may be described on the macroscopic level by cross-diffusion equations. The random influence of the environment motivates the introduction of multiplicative noise terms, leading to systems of nonlinear stochastic parabolic equations with generally not positive definite diffusion matrices. The lack of positive definiteness is overcome by exploiting the entropy (or thermodynamic) structure of the model, which means that a priori estimates are derived by the use of nonlinear test functions. The mathematical difficulty is that this excludes usual stochastic Galerkin or semigroup approaches.
In this talk, a new regularization procedure is presented. It is based on a regularization of the entropy variable through an abstract regularization operator due to Krein et al. (1982), which allows us to obtain local strong solutions. Global estimates follow from the entropy structure and Itô's lemma, and the de-regularization limit is performed by using tightness results. The main result is the global existence of martingale solutions to stochastic cross-diffusion systems with entropy structure and suitable entropy-noise interactions. This is joint work with M. Braukhoff (Germany) and F. Huber (Vienna).