A new Gel'fand triple exp (S)1 ⊆ (L)2 ⊆ exp (S)-1 is constructed as extension of the known Kondratiev one (S)1 ⊆ (L)2 ⊆ (S)-1. Expansion theorems for generalized stochastic processes considered as elements of the spaces [Formula: see text] and [Formula: see text] are derived. This series expansion is used for solving a class of evolution stochastic differential equations. The Wick product is developed on the spaces exp (S)-1, [Formula: see text] and [Formula: see text]. The series expansion of generalized stochastic processes is used for solving a class of nonlinear stochastic differential equations by means of Wick products.

We treat the stochastic Dirichlet problem L♦u = h + ∇ f in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space W 1,2 0 into the Kondratiev space (S) −1 . The operator L is assumed to be strictly elliptic in divergence form L♦u = ∇(A♦∇u + b ♦u) + c♦∇u + d♦u. Its coefficients: the elements of the matrix A and of the vectors b , c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by ♦, is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to L♦u = h + ∇ f .

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We consider three fundamental equations of the Malliavin calculus: the equation involving the Malliavin derivative, the Skorokhod integral and the Ornstein -Uhlenbeck operator of order k, k [ N. These equations provide a complete characterization of the domain and range of the aforementioned operators. Applying the chaos expansion method in white noise spaces we solve these equations and obtain an explicit form of the solutions in the space of Kondratiev generalized stochastic processes.