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

For a stochastic cross-diffusion population system with superquadratic transition rate, we show that a global martingale solution exists. We rely on the Galerkin approximation scheme to derive the sequence of approximated solutions. We apply the It$\rm\hat{o}$ 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. We notice that in the uniform estimation process, we have to introduce an auxiliary sequence. The estimation of the approximated sequence has to be derived on the estimation of the auxiliary sequence, which is the key idea of this work.