Algebra of Generalized Stochastic Processes and the Stochastic Dirichlet Problem

摘要

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{sup}(2,2);(S){sub}(-1)). This is done by using the chaos expansion form of generalized stochastic processes regarded as linear continuous mappings from the Sobolev space (W{sub}0){sup}(k,2) 0 into the Kondratiev space (S){sub}(-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 = {nabla} ·(A · {nabla}u) + c · {nabla}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.