Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior

Abstract

We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function.