On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle

Abstract

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 .

On the Generalized Stochastic Dirichlet Problem—Part II: Solvability, Stability and the Colombeau Case

“…In [13] we proved the weak maximum principle: If L♦u is positive in a weak sense, then u attains its supremum on the boundary of I. This implies that the homogeneous Dirichlet problem L♦u = 0 has only a trivial solution.…”

“…This paper is a conclusion of the first part of this article [13]. We consider the Dirichlet problem…”

“…Especially, one does not need to take care that the PDE-solution belongs to the domain of the inverse Hermite transform. We will also use the series expansion method as we did in [12,13] (also known as the propagator method in [10]) to obtain the coefficients in the chaos expansion of the solution.…”

“…The paper is organized in the following manner: In Section 2 we provide a basic overview of the notation and results obtained in [13]. Section 3 is devoted to solving the non-homogeneous Dirichlet problem (1).…”

Elliptic equations of higher stochastic order

“…The white noise approach exploits the built-in set of stochastic spaces, such as Hida or Kondratiev spaces [11,12], or even larger exponential spaces [16]. The traditional approach [17,20,21], etc., has to select a stochastic space and then to study the largest possible class of equations admitting a solution in that space. The difference of our approach is that we select the stochastic space that is in some sense optimal for the particular equation under consideration.…”