Global martingale solutions for a stochastic population cross-diffusion system

TL;DR

This paper proves the existence of global nonnegative martingale solutions for a stochastic cross-diffusion system modeling multiple interacting populations, incorporating environmental noise and a quadratic entropy structure.

Contribution

It introduces a novel approach to establish solutions for stochastic cross-diffusion systems with non-symmetric, non-positive definite diffusion matrices using energy estimates and advanced probabilistic methods.

Findings

Existence of solutions under general conditions

Solutions maintain nonnegativity

Applicable to systems with complex diffusion structures

Abstract

The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brze\'zniak and co-workers, and Jakubowski's generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia's truncation method due to Chekroun, Park, and Temam.