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Published in:
2015 | OriginalPaper | Chapter
The L p -Poincaré Inequality for Analytic Ornstein–Uhlenbeck Semigroups
Author : Jan van Neerven
Published in: Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics
Publisher: Springer International Publishing
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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 WH 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.