The L ^p-Poincaré Inequality for Analytic Ornstein–Uhlenbeck Semigroups

Abstract

Consider the linear stochastic evolution equation

$$ dU\left( t \right) = AU\left( t \right)dt + dW_H \left( t \right),\,t \geqslant 0, $$

where A generates a C₀-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∞ we prove that if the associated Ornstein–Uhlenbeck semigroup is analytic and has compact resolvent, then the Poincaré inequality

$$ \left\| {f - \bar f} \right\|_{L_p \left( {E,\,\mu _\infty } \right)} \, \leqslant \left\| {D_H \,f} \right\|_{L_p \left( {E,\,\mu _\infty } \right)\,} $$

holds for all 1 < p < ∞. Here f denotes the average of f with respect to µ∞ and DH the Fréchet derivative in the direction of H.

Mathematics Subject Classification (2010). Primary 47D07; Secondary: 35R15, 35R60.