Nonsymmetric Ornstein–Uhlenbeck Semigroups in Banach Spaces

Abstract

Let E be a separable real Banach space and let Q # L(E*, E) be positive and symmetric. Let S=[S(t)] t 0 be a C 0 -semigroup on E We study the relations between the reproducing kernel Hilbert spaces associated with the operators Q t := t 0 S(s) QS*(s) ds. Under the assumption that these operators are the covariances of centered Gaussian measures + t on E, we also study equivalence + t t+ s for different values of s and t, and we calculate their Radon Nikodym derivatives. Academic Press INTRODUCTIONIn this paper we investigate the reproducing kernel Hilbert spaces and Gaussian measures associated with a nonsymmetric Ornstein Uhlenbeck semigroup on a separable real Banach space E. This study is usually carried out in a Hilbert space setting, and one of the motivations of this paper was to see to what extent the theory can be extended to the Banach space setting.The main difference between the Banach space and the Hilbert space situation is that the covariance operator of a Gaussian measure on a Banach space E is a positive symmetric operator Q (the precise definitions are given in Section 1) from the dual E* into E, rather than an operator on E. Thus, in contrast to the Hilbert space situation, it is no longer possible to represent the reproducing kernel Hilbert space H associated with Q as H=Im Q 1Â2. When working in a Banach space setting, any reference to the operator Q 1Â2 therefore has to be avoided. This turns out article no. FU973237 495

On the eigenfunctions of the complex Ornstein–Uhlenbeck operators

“…The other way is that by defining the complex variable creation operator and annihilation operator in the complex Hilbert space L 2 C (µ), we verify that Hermite-Laguerre-Ito polynomials can be generated iteratively by the complex creation operator acting on the constant 1 (see Definition 2.4). Those approaches give us some deeper and richer understanding of the 1-dimensional complex-valued Ornstein-Uhlenbeck processes and the "nonsymmetric" stochastic analysis [3,20,22,23]. The concrete calculations are given in section 2.…”

On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

“…Then K h ∈ L p (E, µ ∞ ) for all 1 < p < ∞, and by a second quantization argument (see [4,12]) we see that P (t)K h = K S * ∞ (t)h , h ∈ H ∞ , t 0, first in L 2 (E, µ ∞ ) and then also in L p (E, µ ∞ ) by consistency. By an analytic continuation argument, this implies that…”

“…The reproducing kernel Hilbert space (RKHS) associated with Q will be denoted by H and the canonical inclusion mapping H ֒→ E by i. We refer to [12] for more details. Whenever this is convenient, we shall identify H with its image i(H) in E.…”

SECOND QUANTIZATION AND THE Lp-SPECTRUM OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS

“…By (2.5) we have Q ∞ = i µ • i * µ . The key fact is the following result, due to Chojnowska-Michalik and Goldys [9] under some additional assumptions; the present formulation was given in [26 By complexification, S µ,C is a strongly continuous semigroup of contractions on H µ,C , and upon identifying L 2 (E, µ) with Γ s (H µ,C ) as explained in the previous section we have the following representation of P 2 , again due to ChojnowskaMichalik and Goldys [9]; see also [26,Theorem 6.12 …”