Global existence of martingale solutions to stochastic keller-segel system with degenerate diffusion

In this paper, we study the stochastic degenerate Keller-Segel system perturbed by linear multiplicative noise in a bounded domain $\mathcal{O}$. We establish the global existence of martingale solutions for this model with any nonnegative initial data in $H_{2}^{-1}(\mathcal{O})$. The main challenge in proving the existence of solutions arises from the degeneracy of the porous media diffusion and the lack of coercivity in the nonlinear chemotactic term. To overcome these difficulties, we construct a solution operator and apply the Schauder fixed point theorem within the variational framework.