The Stochastic Dirichlet Problem Driven by the Ornstein–Uhlenbeck Operator: Approach by the Fredholm Alternative for Chaos Expansions

In this article, we prove the Fredholm alternative theorem for mappings defined on spaces of generalized stochastic processes given by their Wiener–It chaos expansion form. We apply the result to solve the stochastic Dirichlet problem with a perturbation term driven by the Ornstein–Uhlenbeck operator.