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Global martingale solutions for a stochastic population cross-diffusion system

Author

Listed:
  • Dhariwal, Gaurav
  • Jüngel, Ansgar
  • Zamponi, Nicola

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.

Suggested Citation

  • Dhariwal, Gaurav & Jüngel, Ansgar & Zamponi, Nicola, 2019. "Global martingale solutions for a stochastic population cross-diffusion system," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3792-3820.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:10:p:3792-3820
    DOI: 10.1016/j.spa.2018.11.001
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    References listed on IDEAS

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    1. Hofmanová, Martina & Zhang, Tusheng, 2017. "Quasilinear parabolic stochastic partial differential equations: Existence, uniqueness," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3354-3371.
    2. Brzezniak, Zdzislaw & Gatarek, Dariusz, 1999. "Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 187-225, December.
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    Cited by:

    1. Ying Yu & Yahui Chen & You Zhou, 2023. "Cross-Diffusion-Induced Turing Instability in a Two-Prey One-Predator System," Mathematics, MDPI, vol. 11(11), pages 1-12, May.

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