Path decomposition of a spectrally negative L\'evy process, and local time of a diffusion in this environment

TL;DR

This paper investigates the behavior of local time and favorite sites of a transient diffusion in a spectrally negative Lévy environment, revealing convergence properties related to the process's valleys and minima.

Contribution

It introduces a novel analysis of h-valleys and minima in spectrally negative Lévy processes, establishing convergence results for local time and favorite site distributions.

Findings

Renormalized h-minima sequence converges to Poisson process jumps

Distribution of supremum of local time converges in distribution

Characterization of diffusion behavior in Lévy potential environments

Abstract

We study the convergence in distribution of the supremum of the local time and of the favorite site for a transient diffusion in a spectrally negative L\'evy potential. To do so, we study the h-valleys of a spectrally negative L\'evy process, and we prove in partiular that the renormalized sequence of the h-minima converges to the jumping times sequence of a standard Poisson process.