Lévy-Ornstein-Uhlenbeck transition semigroup as second quantized operator
type: Item
instance of: scholarly article
title: Lévy-Ornstein-Uhlenbeck transition semigroup as second quantized operator
zbMATH Open document ID: 1231.47041
published in: Journal of Functional Analysis
publication date: 2011-05-20T00:00:00Z
review text: Let \(\mu\) be an invariant measure for the transition semigroup \((P_t)\) of the Markov family defined by the Ornstein-Uhlenbeck type equation \(dX=AX\,dt+dL\) on a Hilbert space \(E\) driven by a Lévy process \(L\). The author shows that, for any \(t\geq 0\), \(P_t\) considered on \(L^2(\mu )\) is a second quantized operator on the Poisson Fock space of \(e^{At}\). A similar interpretation for stochastic equations driven by the Wiener process was given by \textit{A. Chojnowska-Michalik} and \textit{B. Goldys} [J. Math. Kyoto Univ. 36, No.~3, 481--498 (1996; Zbl 0882.47013)]. For the case where \(E=\mathbb R\) and \(L\) is an \(\alpha\)-stable process, \(0<\alpha <2\), the above result implies that the transition semigroup of the one-dimensional Lévy-Ornstein-Uhlenbeck process is neither symmetric nor compact.
reviewed by: Anatoly N. Kochubei
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 5898816
zbMATH Keywords: Lévy-Ornstein-Uhlenbeck process
zbMATH Keywords: Poisson Fock space
zbMATH Keywords: second quantised operator
author: Szymon Peszat
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.03.002
OpenAlex ID: W2028457335
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DOI: 10.1016/J.JFA.2011.03.002
description: scientific article; zbMATH DE number 5898816