A symmetry result for the ornstein-uhlenbeck operator

In 1978 E. De Giorgi formulated a conjecture concerning the onedimensional symmetry of bounded solutions to the elliptic equation u = F0(u), which are monotone in some direction. In this paper we prove the analogous statement for the equation uδ hx;ruiu = F0(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 an similar result in innite dimensions by a limit procedure.