Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

Let μ be an invariant measure for the transition semigroup (Pt ) of the Markov family defined by the Ornstein–Uhlenbeck type equation dX = AXdt + dL on a Hilbert space E, driven by a Lévy process L. It is shown that for any t 0, Pt considered on L2(μ) is a second quantized operator on a Poisson Fock space of eAt. From this representation it follows that the transition semigroup corresponding to the equation on E = R, driven by an α-stable noise L, α ∈ (0, 2), is neither compact nor symmetric. © 2011 Elsevier Inc. All rights reserved.