Stochastic Diffusion Driven by Internal and External Brownian Motions

The purpose of this paper is to introduce a stochastic version of Fokas' unified transform formula for evolution PDEs on domains with boundary. The unified transform method, also known as Fokas method, was originally developed for deterministic initial-boundary value problems and has, so far, been used primarily in deterministic settings. The modifications introduced here lead to stochastic representation formulas that can be used to define mild solutions of stochastic initial-boundary value problems and to perform regularity analysis in rough fractional Sobolev spaces, below the algebra threshold. To present the main ideas, we consider the canonical model of linear and nonlinear diffusion equations with additive noise and stochastic boundary conditions in their most elementary form. We assume that the evolution is subject both to external forcing prescribed through Dirichlet-type Brownian motion and to internal additive forcing whose temporal component is also Brownian.