Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

AbstractLet μ be an invariant measure for the transition semigroup (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⩾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