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The existence of global nonnegative martingale solutions to a stochasticcross-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 Lipschitzproperty. We introduce a sequence of diffusion matrices satisfying theLipschitz property, and apply the existence and uniqueness theorem for thestochastic differential equation to establish the existence of a unique strongsolution result. The entropy method adapting to the deterministiccross-diffusion system may not be able to provide strong enough uniformestimates for a tightness proof in the stochastic environment. Under astrengthened coefficients of the diffusion matrix condition, an application ofthe It$\rm\hat{o}$ formula to a linear transformation between variables issuffice to provide us with strong enough uniform estimates. After the tightnessproperty be proved based on the estimation, a space changing result be used toconfirm the limit is a weak solution to the cross-diffusion system. Nonnegativeproperty for the martingale solution is proved via a standard Stampacchia-typeargument.

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