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

TL;DR

This paper constructs non-negative martingale solutions for a stochastic porous medium equation in one dimension that exhibit sticky behavior at zero, using a stochastic Galerkin method and weak convergence techniques.

Contribution

It introduces a novel construction of solutions with sticky boundary behavior for the stochastic porous medium equation using a combination of spatial discretization and probabilistic methods.

Findings

Successfully constructed martingale solutions with sticky zero behavior

Developed uniform moment estimates for discretized systems

Applied weak convergence and occupation time formulas in the analysis

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.