Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

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