On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes

Abstract

We show that shift Harnack type inequalities (in the sense of Wang (2014)) are preserved under Bochner's subordination. The proofs are based on two types of moment estimates for subordinators. As a by-product we establish moment estimates for general Lévy processes.

Transition probabilities of Lévy‐type processes: Parametrix construction

“…Secondly, it is known that the behaviour of the characteristic exponent at zero is closely related to the existence and growth of fractional moments ( | | ) , > 0, cf. [11,Section 3] and [35]. Since the existence of exponential and fractional moments allows us to draw direct conclusions on the decay of the transition density, this illustrates that the growth behaviour and the domain of analyticity of the characteristic exponent play an important role for heat kernel estimates.…”

Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs

“…isotropic α-stable, relativistic stable, tempered stable and layered stable Lévy processes. The proof relies on the so-called Itô-Tanaka trick which relates the time average t 0 b(s, X s ) ds of the solution (X t ) t≥0 to (1) with the solution u to the Kolmogorov equation (2) ∂ t u(t, x) + A x u(t, x) + b(t, x) · ∇ x u(t, x) = −b(t, x);…”

“…here A x denotes the infinitesimal generator of the driving Lévy process (L t ) t≥0 acting with respect to the space variable x. The key step is to prove the existence of a solution to (2) which is sufficiently regular and satisfies certain Hölder estimates. The required regularity of u depends on the regularity of b and the behaviour of the Lévy measure at 0.…”

Log‐Harnack inequalities for Markov semigroups generated by non‐local Gruschin type operators

“…Proof of Example By the self‐similar property of α‐stable subordinators, one has On the other hand, it is clear that Since for all , we get that for all Here, means that for some constant and all . This, together with [, Theorem 3.8 (a)] (or [, Theorem 2.1 b)]), yields that for some constant …”