Global martingale solutions to a stochastic cross-diffusion population system between superlinear and subquadratic transition rate case (2203.14311v6)

Abstract: The existence of global nonnegative martingale solutions to a stochastic cross-diffusion population system with the power of the transition rate $1<s<2$ is shown. When $1<s<2$, the diffusion matrix does not satisfy the Lipschitz property. We introduce a sequence of diffusion matrices satisfying the Lipschitz property, and apply the existence and uniqueness theorem for the stochastic differential equation to establish the existence of a unique strong solution result. The entropy method adapting to the deterministic cross-diffusion system may not be able to provide strong enough uniform estimates for a tightness proof in the stochastic environment. Under a strengthened coefficients of the diffusion matrix condition, an application of the It$\rm\hat{o}$ formula to a linear transformation between variables is suffice to provide us with strong enough uniform estimates. After the tightness property be proved based on the estimation, a space changing result be used to confirm the limit is a weak solution to the cross-diffusion system. Nonnegative property for the martingale solution is proved via a standard Stampacchia-type argument.