A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications

The purpose of this paper is to study a boundary reaction problem on the space X×ℝ, where X is an abstract Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation (-Δγ)su = f(u), with s ∈ (0, 1), where (-Δγ)s denotes a fractional power of the Ornstein-Uhlenbeck operator, and we prove that for any s ∈ (0, 1) monotone solutions are one-dimensional