Asymptotic behavior of densities of unimodal convolution semigroups

TL;DR

This paper derives asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups on Euclidean space, assuming their Lévy–Khintchine exponents are regularly varying with index between 0 and 2.

Contribution

It provides the first detailed asymptotic analysis of densities for a broad class of unimodal convolution semigroups under regular variation assumptions.

Findings

Established asymptotic formulas for transition densities.

Extended understanding of unimodal convolution semigroups.

Connected Lévy–Khintchine exponents with density behavior.

Abstract

We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^d$ under the assumption that its L\'{e}vy--Khintchine exponent is regularly varying of index between $0$ and $2$.