On the Cauchy problem for non-local Ornstein–Uhlenbeck operators

Abstract

We study the Cauchy problem involving non-local Ornstein-Uhlenbeck operators in finite and infinite dimensions. We prove classical solvability without requiring that the Lévy measure corresponding to the large jumps part has a first finite moment. Moreover, we determine a core of regular functions which is invariant for the associated transition Markov semigroup. Such a core allows to characterize the marginal laws of the Ornstein-Uhlenbeck stochastic process as unique solutions to Fokker-Planck-Kolmogorov equations for measures.

Heat kernel of anisotropic nonlocal operators

“…We note in passing that in principle [37] allows to handle anisotropic kernels, but precise upper estimate of the resulting heat kernel are non-trivial to obtain from the series representation given there. The reader interested in probabilistic methods may consult further results and references in [12,36,37,39,42,45].…”

Functional inequalities for some generalised Mehler semigroups

“…In [5,Theorem 5.2], see also [39,Remark 5.11], it is shown that FC 2 A (X), under Hypothesis 2.1, is a core for the generator of P t in C b (X) equipped with the mixed topology. Recall that a core of an operator A : D(A) ⊆ X → X is a subspace C ⊆ D(A) which is dense in D(A) with respect to the graph norm…”

“…Proof. We point out that FC 2 A (X) is invariant with respect to P t (see [5,Theorem 5.2] and [39,Remark 5.11]), so to conclude we just need to show that FC 2 A (X) is contained in the domain of the generator L in L 2 (X, σ) and that it is dense in L 2 (X, σ).…”

“…, y χ B1 (y)]M (dy), (1.3) on regular functions. We refer to Section 2 for a more detailed explanation, to [38] for a general introduction to these topics, to [5,6,12,15,20,28,29,39,40,41,45] as more specific basic references to generalised Mehler semigroups and to the very recent [32] and the reference therein for an updated account on the regularity theory, which we do not discuss here.…”

Pathwise uniqueness for stochastic evolution equations with Hölder drift and stable Lévy noise

“…Although the Ornstein-Uhlenbeck R t is not a C 0 -semigroup on C b (H), we can define its generator (L, D(L)) by point-convergence topology (see, for instance, [9,12]) and its resolvent R(λ, L) of L is given by…”