This paper addresses the existence and uniqueness of strong solutions to stochastic porous media equations dX−ΔΨ(X) dt=B(X) dW(t) in bounded domains of ℝd with Dirichlet boundary conditions. Here Ψ is a maximal monotone graph in ℝ×ℝ (possibly multivalued) with the domain and range all of ℝ. Compared with the existing literature on stochastic porous media equations, no growth condition on Ψ is assumed and the diffusion coefficient Ψ might be multivalued and discontinuous. The latter case is encountered in stochastic models for self-organized criticality or phase transition.
"Existence of strong solutions for stochastic porous media equation under general monotonicity conditions." Ann. Probab. 37 (2) 428 - 452, March 2009. https://doi.org/10.1214/08-AOP408