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 (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.
This article was supported by the project Functional Analysis Methods, ODEs, and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.