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Gradient Estimates and Domain Identification for Analytic Ornstein-Uhlenbeck Operators

  • Jan Maas
  • Jan van Neerven
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 80)

Abstract

Let P be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem
$$ dU(t) = AU(t)\,dt + dW_{H}(t), $$
where A is the generator of a C 0-semigroup S on a Banach space E, H is a Hilbert subspace of E, and W H is an H-cylindrical Brownian motion. Assuming that S restricts to a C 0-semigroup on H, we obtain L p -bounds for D H P(t). We show that if P is analytic, then the invariance assumption is fulfilled. As an application we determine the L p -domain of the generator of P explicitly in the case where S restricts to a C 0-semigroup on H which is similar to an analytic contraction semigroup. The results are applied to the 1D stochastic heat equation driven by additive space-time white noise.

Keywords

Ornstein-Uhlenbeck operators gradient estimates domain identification. 

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany
  2. 2.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands

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