On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

Abstract

We study Hilbert space valued Ornstein-Uhlenbeck processes (Y (t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C 0 semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.

Regularity of Ornstein–Uhlenbeck Processes Driven by a Lévy White Noise

“…where x ∈ E and A is assumed to be an infinitesimal generator of a C 0 -semigroup S = (S(t)) t≥0 on a Banach space E. Such equations have been investigated by many authors in a Hilbert space setting, for example by Chojnowska-Michalik in [15], Fuhrman and Röckner in [21], Lescot and Röckner in [28] and Applebaum in [4]. In the most interesting cases the semigroup S is analytic and the space E is a subspace of the space H, i.e.…”

Uniqueness for solutions of Fokker–Planck equations related to singular SPDE driven by Lévy and cylindrical Wiener noise

“…in [FR00], [SS01], [LR02], [App07] (see also the references therein). In [LR02], the restriction of the infinitesimal generator U of (S t ) to a suitable space of test functions W A (see Subsection 1.5 below) has been identified as U ψ(x) = H i Aξ, x − λ(ξ) · e i ξ,x F −1 ψ( · ) (dξ) , ψ ∈ W A , where F −1 denotes the inverse Fourier transform.…”

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

“…Pseudo-differential operator representations of L have been obtained in [32], and in Proposition 4.1 of [4]. The measures (µ t , t ≥ 0) have an interesting property, which will play an important role in the sequel.…”