Global martingale solutions for a stochastic population cross-diffusion system
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.
Cite
@article{arxiv.1806.01124,
title = {Global martingale solutions for a stochastic population cross-diffusion system},
author = {Gaurav Dhariwal and Ansgar Jüngel and Nicola Zamponi},
journal= {arXiv preprint arXiv:1806.01124},
year = {2020}
}