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 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 (-Delta(gamma))(s)u = f(u), with s is an element of (0, 1), where (-Delta(gamma))(s) denotes a fractional power of the Ornstein-Uhlenbeck operator, and we prove that for any s is an element of (0, 1) monotone solutions are one-dimensional.