The Ornstein-Uhlenbeck semigroup in exterior domains

Let Omega be an exterior domain in R^n. It is shown that Ornstein-Uhlenbeck operators L generate C_0-semigroups on L^p(Omega) for p in (1, \infty) provided Omega is smooth. The method presented also allows to determine the domain D(L) of L and to prove L^p-L^q smoothing properties of e^{tL}. If Omega is only Lipschitz, results of this type are shown to be true for p close to 2