Global martingale solutions for a stochastic Shigesada-Kawasaki-Teramoto population model

TL;DR

This paper proves the existence of global martingale solutions for a stochastic cross-diffusion population model with multiple species, using entropy methods and approximation techniques.

Contribution

It establishes the existence of solutions for a complex stochastic population model with non-symmetric, non-positive diffusion matrices, extending mathematical understanding of such systems.

Findings

Existence of global nonnegative martingale solutions is proven.

The model handles arbitrary number of species with segregation dynamics.

Standard methods are excluded due to the diffusion matrix properties.

Abstract

The existence of global nonnegative martingale solutions to a cross-diffusion system of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the segregation dynamics of populations with an arbitrary number of species. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods. Instead, the existence proof is based on the entropy structure of the model, approximated by a Wong-Zakai argument, and on suitable higher moment estimates and fractional time regularity. In the case without self-diffusion, the lack of regularity is overcome by carefully exploiting the entropy production terms.