Path Decomposition of a Spectrally Negative L\'evy Process, and Local Time of a Diffusion in This Environment

#### G. Vechambre

2018, v.24, Issue 4, 563-668

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 particular that the renormalized sequence of the $h$-minima converges to the jumping times sequence of a standard Poisson process.

Keywords: diffusion, random potential, Levy process, renewal process, local time