Global Martingale Solutions to a Stochastic Superlinear Cross-Diffusion Population System

Abstract

The existence of global martingale solutions to a stochastic superlinear cross-diffusion population system is shown. The diffusion matrix does not satisfy the local Lipschitz property. We have to regularize the diffusion matrix in order to apply the existence and uniqueness theorem. By applying the existence and uniqueness theorem, we derive a sequence of approximated solutions. Then we apply the It$\rm\hat{o}$ formula to a linear transformation between variables to estimate approximated solutions. We can show that the sequence of approximated solutions is tight in a topological space, with its limit a martingale solution of the stochastic cross-diffusion system. Nonnegative property for a martingale solution is proved via a standard Stampacchia-type argument.

Keywords: martingale solutions, tightness criterion, stochastic cross-diffusion population system