Global martingale solutions for a stochastic population cross-diffusion system

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...

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Bibliographic Details
Published in:arXiv.org 2018-06
Main Authors: Dhariwal, Gaurav, Jüngel, Ansgar, Zamponi, Nicola
Format: Article
Language:English
Subjects:
Diffusion
Galerkin method
Hilbert space
Martingales
Matrix methods
Tightness
Online Access:Get full text
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Summary: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 Brzeźniak 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.
ISSN:2331-8422
DOI:10.48550/arxiv.1806.01124