Asymptotic behaviour and estimates of slowly varying convolution semigroups

TL;DR

This paper investigates the long-term behavior and estimates of transition densities for certain stochastic processes with slowly varying characteristics, providing new asymptotic formulas and bounds.

Contribution

It introduces novel asymptotic formulas and estimates for transition densities of isotropic unimodal convolution semigroups with slowly varying Lévy--Khintchine exponents.

Findings

Derived asymptotic formulas for transition densities.

Established new estimates for Green functions.

Analyzed behavior of convolution semigroups with slowly varying exponents.

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 varies slowly. We also derive some new estimates of the transition densities and Green functions.