Exponential decay of measures and Tauberian theorems

TL;DR

This paper investigates how the decay rate of measures on [0,∞) relates to their Laplace transforms, establishing conditions under which exponential decay occurs and exploring heavy tail behavior with applications to non-local equations.

Contribution

It extends Nakagawa's results by linking Laplace transform analyticity to measure decay rates and analyzes measure densities and heavy tails under various assumptions.

Findings

Exponential decay of measures is characterized by the abscissa of convergence.

Under stronger conditions, the density behavior of measures can be deduced from their Laplace transforms.

Heavy tail phenomena are studied with applications to non-local equations.

Abstract

We study behavior of a measure on $[0,\infty)$ by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend the result of Nakagawa from [Nak05]. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy tails and give an application in the theory of non-local equations.