The L^p-Poincar\'e inequality for analytic Ornstein-Uhlenbeck operators

Consider the linear stochastic evolution equation dU(t) = AU(t) + dW_H(t), t\ge 0, where A generates a C_0-semigroup on a Banach space E and W_H is a cylindrical Brownian motion in a continuously embedded Hilbert subspace H of E. Under the assumption that the solutions to this equation admit an invariant measure \mu_\infty we prove that if the associated Ornstein-Uhlenbeck semigroup is analytic and has compact resolvent, then the Poincar\'e inequality \n f - \overline f\n_{L^p(E,\mu_\infty)} \le \n D_H f\n_{L^p(E,\mu_\infty)} holds for all 1<p<\infty. Here \overline f denotes the average of f with respect to \mu_\infty and D_H the Fr\'echet derivative in the direction of H.Comment: Minor correctiopns. To appear in the proceedings of the symposium "Operator Semigroups meet Complex Analysis, Harmonic Analysis and Mathematical Physics", June 2013, Herrnhut, German

L^2-Theory for non-symmetric Ornstein-Uhlenbeck semigroups on domains

We present some new results on analytic Ornstein-Uhlenbeck semigroups and use them to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck semigroups on open domains O to the non-symmetric case. Denoting the generator of the semigroup by L_O, we obtain sufficient conditions in order that the domain Dom(\sqrt{-L_O}) be a first order Sobolev space.Comment: 23 pages, revised version, to appear in J. Evol. Eq. The main change is a correction in Theorem 5.5: the second assertion has been withdrawn due to a gap in the original proo