Global martingale solutions to a stochastic superquadratic Shigesada-Kawasaki-Teramoto type population system

Xi Lin

doi:10.59400/adecp3619

Global martingale solutions to a stochastic superquadratic Shigesada-Kawasaki-Teramoto type population system

Xi Lin Department of Mathematics and Physics, Guangzhou Maritime University, Guangzhou 510725, China

Article ID: 3619

Keywords: global martingale solutions, existence of solutions, Galerkin approximation, auxiliary sequence, tightness criterion, stochastic Shiegesada-Kawasaki-Teramoto (SKT) system, superquadratic cross-diffusion, population dynamics

Abstract

For a stochastic cross-diffusion population system with superquadratic transition rate, we show that a global martingale solution exists. The existence of a global martingale solution proof for a stochastic population system is quite different from the existence of a weak solution proof for a deterministic population system. For deterministic population systems, we apply the entropy method to show a weak solution exists. For stochastic population systems, we rely on the Galerkin approximation scheme to derive the sequence of approximated solutions. We apply the Itbo formula to derive uniform estimates. After the tightness property be proved based on the estimation, a space changing result then be used to confirm the limit is a martingale solution of the cross-diffusion system. In the uniform estimation process, we notice that we have to estimate stochastic processes that are not in the finite dimensional space, otherwise we can not derive strong enough estimation for the tightness proof. We are not able to apply the Itbo formula to stochastic processes that are not finitely dimensional processes. In this situation, we have to introduce an auxiliary sequence. The estimation of the approximated sequence has to be derived on the estimation of an auxiliary sequence, which is the key idea of this work. The nonnegative property for approximated solutions has been shown by the standard Stampacchia method.

Published

2025-09-30

How to Cite

Lin, X. (2025). Global martingale solutions to a stochastic superquadratic Shigesada-Kawasaki-Teramoto type population system. Advances in Differential Equations and Control Processes, 32(3). https://doi.org/10.59400/adecp3619

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Vol. 32 No. 3 (2025)

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

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