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 (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.

Keywords:

• Chaos expansion
• Colombeau algebra
• Elliptic linear differential operator
• Generalized expectation
• Generalized random process
• Wick product

Mathematics Subject Classification:

• 60G20
• 60H30
• 46F30
• 60H40
• 46N30

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.