Solving Stochastic Eigenvalue Problem of Wick Type

In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Ito chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm alternative. To reduce the computational complexity of this system, we shall use a decomposition method using the Wiener-Ito chaos expansion. Once the approximation of the solution is performed using the finite element method for example, the statistics of the numerical solution can be easily evaluated. Keywords—Eigenvalue problem, Wick product, SPDEs, finite element, Wiener-Ito chaos expansion.