Global martingale solutions for a stochastic population cross-diffusion system |
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Institution: | 1. Department of Statistics and Probability, Michigan State University, USA;2. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” — Università degli studi di Napoli “Federico II”, Italy;1. Université de Cergy-Pontoise, UMR AGM8088, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise CEDEX, France;2. Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland;3. Federal University of Rio Grande do Sul, Mathematics Institute, Av. Bento Gonçalves, 9500 Porto Alegre, RS, Brazil;4. Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, 03225 Vilnius, Lithuania |
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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 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. |
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Keywords: | Shigesada–Kawasaki–Teramoto model Population dynamics Martingale solutions Tightness Skorokhod–Jakubowski theorem Stochastic maximum principle |
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