Global martingale solutions for a stochastic population cross-diffusion system. (English) Zbl 1422.35186
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 Z. Brzeźniak et al. [AMRX, Appl. Math. Res. Express 2013, No. 1, 1–33 (2013; Zbl 1272.60041)], and A. Jakubowski’s [Teor. Veroyatn. Primen. 42, No. 1, 209–216 (1997; Zbl 0923.60001)] generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia’s truncation method due to M. D. Chekroun et al. [J. Differ. Equations 260, No. 3, 2926–2972 (2016; Zbl 1343.35247)].
MSC:
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35K57 | Reaction-diffusion equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 92D25 | Population dynamics (general) |
Keywords:
Shigesada-Kawasaki-Teramoto model; population dynamics; martingale solutions; tightness; Skorokhod-Jakubowski theorem; stochastic maximum principleReferences:
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