\(L ^{2}\)-theory for non-symmetric Ornstein-Uhlenbeck semigroups on domains. (English) Zbl 1302.60094
Summary: We prove that the mild solution of the stochastic evolution equation \({\text{d}X(t) = AX(t)\,\text{d}t + \text{d}W(t)}\) on a Banach space \(E\) has a continuous modification if the associated Ornstein-Uhlenbeck semigroup is analytic on \(L ^{2}\) with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck semigroups on domains \({\mathcal{O} \subseteq E}\) to the non-symmetric case. Denoting the generator of the Ornstein-Uhlenbeck semigroup by \({L_\mathcal{O}}\), we obtain sufficient conditions in order that the domain of \({\sqrt{-L_\mathcal{O}}}\) be a first-order Sobolev space.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
Keywords:
analytic Ornstein-Uhlenbeck semigroups; Riesz transforms on domains; domain identification; Poincaré inequality; \(H ^{\infty }\)-functional calculusReferences:
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