# L{\'e}vy processes: concentration function and heat kernel bounds

## TL;DR

This paper studies the densities of Lévy processes, establishing equivalences between common conditions on their characteristic exponents and the behavior of their densities over time, along with lower bounds under mild assumptions.

## Contribution

It reveals the equivalence of typical conditions on characteristic exponents with the maximum density behavior and provides qualitative lower estimates for densities.

## Key findings

- Equivalence of conditions on characteristic exponents and density maxima
- Qualitative lower bounds for densities under mild assumptions
- Insights into heat kernel bounds for Lévy processes

## Abstract

We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable. We also prove qualitative lower estimates under mild assumptions on the corresponding jump measure and the characteristic exponent.