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

TL;DR

This paper proves a Poincaré inequality for analytic Ornstein-Uhlenbeck semigroups in Banach spaces under certain compactness and analyticity conditions, extending functional inequalities in stochastic evolution equations.

Contribution

It establishes a new Poincaré inequality for Ornstein-Uhlenbeck operators in Banach spaces with invariant measures, assuming analyticity and compact resolvent conditions.

Findings

Poincaré inequality holds for all 1<p<∞ in this setting.

Analyticity and compact resolvent imply the inequality.

Results extend functional inequalities to infinite-dimensional stochastic systems.

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 \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.