Gradient estimates and domain identification for analytic Ornstein-Uhlenbeck operators

2009/11/23 by Jan Maas, Maas, Jan, Jan van Neerven +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #42B25 #46E35 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Primary: 47D07 #Probability (math.PR) #Secondary: 35R15 #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.0911.4336

openalex publication_date 2009/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (P(t)) 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 C₀-semigroup (S(t)) on a Banach space E, H is a Hilbert subspace of E, and (W_H(t)) is an H-cylindrical Brownian motion. Assuming that (S(t)) restricts to a C₀-semigroup on H, we obtain L^p-bounds for the gradient D_H P(t). We show that if (P(t)) is analytic, then the invariance assumption is fulfilled. As an application we determine the L^p-domain of the generator of (P(t)) explicitly in the case where (S(t)) restricts to a C₀-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.

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