Gradient Estimates and Domain Identification for Analytic Ornstein-Uhlenbeck Operators

Abstract

Let P be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problemwhere 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.Here i : H ֒→ E is the inclusion mapping. A necessary and sufficient condition for the existence of a weak solution is that the operators I t : L 2 (0, t; H) → E, I t g := t 0 S(s)ig(s) ds, 1991 Mathematics Subject Classification. 47D07 (35R15, 42B25, 60H15).

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

“…The proof relies on some facts that have been proved in [25,26]. We start by observing that if Assumptions 2.1 and 2.3 hold, then the semigroup P (t) := P (t) ⊗ S * H (t) extends to a bounded analytic C 0 -semigroup on L p (E, µ ∞ ; H), 1 < p < ∞.…”

Ornstein–Uhlenbeck semigroups in infinite dimension

“…This allows to prove that T ( t ) is smoothing along H , by arguments similar to the ones that led to (3.3)(ii). See refs [19, Sect. 2] and [18] for representation formulae and estimates for any order H -derivatives of T ( t ) f when f ∈ C b ( X ).…”

“…In addition, in the analytic case, the semigroup e tA maps H into itself, and the semigroup S H ( t ) is a strongly continuous, bounded analytic semigroup on H , see ref. [19, Thm. 3.3].…”

Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term

“…The following extension to L p (µ), p > 1, can be found in Section 3 of [3] (see also [2] and [20]).…”