Transition densities of subordinators of positive order

Abstract

A. We prove existence and asymptotic behavior of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if additional upper scaling condition on the Laplace exponent is imposed.

Transition densities of spectrally positive Lévy processes

“…In the non-symmetric case, a similar right tail decay is displayed by transition densities of subordinators (see e.g. [7]). This is also the case for spectrally one-sided Lévy processes, as the following lemma states.…”

“…Proceeding exactly as in the proof of [7,Proposition 4.15] Thus, by Corollary 2.6, for all t ∈ (0, 1/Φ(x 0 )) and x ∈ (0, x 1 ) such that xϕ −1 (1/t) > 1,…”

“…Brownian motion with an independent subordinator, however, is included. For results concerning heat kernel estimates for subordinators we refer the reader to [7] and the references therein.…”

“…Recently, estimates for subordinators in a general setting were obtained (see e.g. [3,4,7]). Still, we are not aware of any article which would treat transition densities of general spectrally one-sided Lévy process of unbounded variation in a comprehensive way.…”

Hitting probabilities for Lévy processes on the real line

“…Apart from its own value, they can be used, together with heat kernel estimates ( [13]) for instance for estimation of the Hausdorff dimension of the inverse images of Lévy processes (see [21]). We also remark that although our main object to operate with is the real part of the characteristic exponent, one can work with the tail of the Lévy measure instead, since in view of [10,Proposition 3.8], scaling property of the latter implies scaling of the former.…”