Fluctuation Correction and Global Solutions for the Stochastic Shigesada-Kawasaki-Teramoto System via Entropy-Based Regularization

Abstract

We study a stochastic extension of the n-species Shigesada-Kawasaki-Teramoto (SKT) cross-diffusion system, in which a multiplicative noise term accounts for fluctuation corrections to the mean-field dynamics arising from a finite underlying population. The noise we consider is motivated by, but not rigorously derived from, the particle-system approximation of Chen, Daus, Holzinger, and J"ungel; it only partially respects the entropy (gradient-flow) structure of the deterministic system, which is the main source of the technical difficulties addressed in this paper. For the resulting system of stochastic partial differential equations (SPDEs), we establish the existence of nonnegative, global, weak martingale solutions, under a detailed-balance condition on the diffusion coefficients and a smallness condition relating the noise intensity to these coefficients. This smallness condition amounts to a sufficiently large population size N (equivalently, small fluctuation strength 1/N): the existence theory thus operates in the perturbative regime around the deterministic mean-field limit, and for fixed coefficients it is always satisfiable once N is large enough.

arXiv admin note: text overlap with arXiv:2202.12602 (same author) Derivation section split off, becomes companion paper