  The existence of a global martingale solution to a cross-diffusion system
with multiplicative Wiener noise in a bounded domain with no-flux boundary
conditions is shown. The model describes the dynamics of population densities
of different species due to segregation cross-diffusion effects. The diffusion
matrix is generally neither symmetric nor positive semidefinite. This
difficulty is overcome by exploiting the Rao entropy structure. The existence
proof uses a stochastic Galerkin method, uniform estimates from the Rao entropy
inequality, and the Skorokhod--Jakubowski theorem. Furthermore, an exponential
equilibration result is proved for sufficiently small Lipschitz constants of
the noise by using the relative Rao entropy. Numerical tests illustrate the
behavior of solutions in one space dimension for two and three population
species.
