  In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional
symmetry of bounded solutions to the elliptic equation \Delta u=F'(u), which
are monotone in some direction. In this paper we prove the analogous statement
for the equation \Delta u - <x, Du> =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 an similar
result in infinite dimensions by a limit procedure.
