The L^p-Poincaré inequality for analytic Ornstein-Uhlenbeck operators

Abstract

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 μ_\infty we prove that if the associated Ornstein-Uhlenbeck semigroup is analytic and has compact resolvent, then the Poincaré inequality \n f - \overline f\n_{L^p(E,μ_\infty)} \le \n D_H f\n_{L^p(E,μ_\infty)} holds for all 1<p<\infty. Here \overline f denotes the average of f with respect to μ_\infty and D_H the Fréchet derivative in the direction of H.

Minor correctiopns. To appear in the proceedings of the symposium "Operator Semigroups meet Complex Analysis, Harmonic Analysis and Mathematical Physics", June 2013, Herrnhut, Germany