On subexponentiality of the L\'evy measure of the diffusion inverse local time; with applications to penalizations

TL;DR

This paper investigates the asymptotic behavior of the local time distribution at zero for recurrent linear diffusions, linking it to the subexponential nature of the Lévy measure of the inverse local time, and applies these findings to penalization procedures.

Contribution

It establishes the asymptotic relation between the local time distribution and the Lévy measure for diffusions, providing explicit constants via spectral methods, and extends penalization results to broader processes.

Findings

Asymptotic behavior of local time distribution characterized

Explicit constants derived for the asymptotics

Generalization of penalization results to new classes of processes

Abstract

For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the L\'evy measure of the inverse local time is subexponential this distribution behaves asymtotically as a multiple of the L\'evy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on $\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.