Lévy-Ornstein-Uhlenbeck transition semigroup as second quantized operator (Q537698)

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.