# Transition densities of subordinators of positive order

## TL;DR

This paper establishes the existence, asymptotic behavior, and bounds for the transition densities of a broad class of subordinators with Laplace exponents satisfying specific scaling conditions, including sharp estimates under additional assumptions.

## Contribution

It provides new results on the transition densities of subordinators with Laplace exponents meeting lower and upper scaling conditions, including sharp estimates in certain cases.

## Key findings

- Proved existence and asymptotic behavior of transition densities.
- Derived lower and upper bounds for the densities.
- Provided sharp estimates under additional scaling conditions.

## Abstract

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. In particular, we cover the case when the (minus) second derivative of the Laplace exponent is a function regularly varying at infinity with regularity index bigger than -2.