Estimates on transition densities of subordinators with jumping density decaying in mixed polynomial orders

Abstract

In this paper, we discuss estimates on transition densities for subordinators, which are global in time. We establish the sharp two-sided estimates on the transition densities for subordinators whose Lévy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on Lévy measures, we also obtain a precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more.

General Law of iterated logarithm for Markov processes: Limsup law

“…Note that a similar result appears in [3, (3.25)] (see also [23, (3.3)]). However, in this paper, we not only give local versions (in the variable r) of that result, but also remove an extra assumption that the local lower scaling index of φ is strictly larger than 1 therein, by using auxiliary functions defined in subsection 3.1 and adopting the ideas from the first and second named authors' previous paper [28,Theorem 3.10].…”

Heat kernel estimates for subordinate Markov processes and their applications

“…Note that β 2 > 1 is allowed. Building upon the results from [22,23], we show several auxiliary results leading to the important estimate (2.24) saying that P(S t ≥ s) ≃ tw(s) for 2φ −1 (1/t) −1 < s < R 1 /2. Sections 3-4 are central to the paper.…”

“…Proof. The proof is similar to that of [23,Lemma 2.1(3)], where the existence of Lévy density is assumed. We give a detailed proof for the reader's convenience.…”

“…On the other hand, by adapting the proof of [23,Lemma 2.3(3)], we can get that, for every r 0 > 0, there exists c 4 = c 4 (r 0 ) > 0 such that…”

“…Suppose that (Poly-R 1 ) holds. Then it follows from the proof of [23,Lemma 2.3(3)] that, for every r 0 > 0, there exists a constant c = c(r 0 ) > 0 such that…”