A Symmetry Result for the Ornstein-uhlenbeck Operator

In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation ∆u = F ′(u), which are monotone in some direction. In this paper we prove the analogous statement for the equation ∆u−〈x,∇u〉u = F ′(u), where the Laplacian is replaced by the Ornstein-Uhlenbeck operator. Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain a similar result in infinite dimensions, by a limit procedure.