Multidimensional Reaction-Diffusion Equations with White Noise Boundary Perturbations

Abstract

In this paper we study multidimensional stochastic reaction-diffusion equations (SRDE's) with white noise boundary data. More precisely, we consider a general SRDE with Robin data to be a white noise field. Because this boundary data is very irregular, we formulate a set of conditions that a random field must satisfy to solve the SRDE. We show that a unique solution exists, and we study the boundary-layer behavior of the solution. This boundary-layer analysis reveals some natural restrictions on the reaction term of the SRDE that ensure that the reaction term does not qualitatively affect the boundary layer. The boundary-layer analysis also leads to the definition of some functional Banach spaces into which are encoded the boundary-layer degeneracies and that would be the natural settings for other analyses of the SRDE of this paper (e.g., large deviations and central limit theorems, approximation theorems).