The Ornstein-Uhlenbeck semigroup in exterior domains

Let Ω be an exterior domain in It is shown that Ornstein-Uhlenbeck operators L generate C₀-semigroups on L^p(Ω) for p ∈ (1, ∞) provided ∂Ω 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 ∂Ω is only Lipschitz, results of this type are shown to be true for p close to 2. Received: 16 December 2004; revised: 4 February 2005