Algebra of Generalized Stochastic Processes and the Stochastic Dirichlet Problem

Abstract

Following the embedding idea of generalized functions into the Colombeau algebra of generalized functions, we construct a new algebra of generalized stochastic processes and denote it 𝒢(W 2,2;(S)−1). This is done by using the chaos expansion form of generalized stochastic processes regarded as linear continuous mappings from the Sobolev space into the Kondratiev space (S)−1. As an application, we prove existence and uniqueness of the solution of the equation Lu = h with given stochastic boundary condition. The operator L is assumed to be strictly elliptic in divergence form Lu = ∇·(A·∇ u) + c·∇ u + du. Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be deterministic Colombeau generalized functions.