On transience of Lévy-type processes

Abstract

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures.for a (transient) Lévy process {L t } t≥0 and any κ > 0, E[L κ B(0,r) ] is either finite or infinite for every r > 0 (see also [Haw77] for the case of symmetric Lévy processes). Accordingly, a (transient) Lévy process {L t } t≥0 is said to be κ-weakly transient if E[L κ B(0,r) ] = ∞ for all r > 0, and κ-strongly transient if E[L κ B(0,r) ] < ∞ for all r > 0. Also, in the same reference, the authors have proved that the above conditions are equivalent to ∞ 0 t κ P(L t ∈ B(0, r))dt = ∞ for all r > 0 and ∞ 0

Decay Estimates in Time for Classical and Anomalous Diffusion

“…In particular, fractional processes are typically recurrent only in dimension 1 and for values of s greater or equal to 1/2 (being transient in dimension 2 and higher, and also in dimension 1 for values of s smaller than 1/2), and this is an important difference with respect to the case of Gaussian processes, which are recurrent in dimensions 1 and 2 (and transient in dimension 3 and higher). See Example 3.5 in [37] and the references therein for a detailed treatment of recurrence and transiency for Lévy-type processes. In this note, in § 2, we present a very simple, and somewhat heuristic, discussion of the recurrence and transiency properties related to the long jump random walks, based on PDE methods and completely accessible to a broad audience.…”