Ray Knight theorems for spectrally negative Lévy processes

Abstract

In this paper, we study the law of the local time processes $(L_T^x(X),x\in \mathbb{R})$ associated to a spectrally negative Lévy process X, in the cases $T={\mathit{\tau }}_a^{+}$, the first passage time of X above $a>0$ and $T=\mathit{\tau }(c)$, the first time it accumulates c units of local time at zero. We describe the branching like structure of local times and Poissonian constructions of them using excursion theory. The presence of jumps for X creates a type of excursions which can contribute simultaneously to local times of levels above and below a given reference point. This fact introduces dependency on local times, causing them to be non-Markovian. Nonetheless, the overshoots and undershoots of excursions will be useful to analyze this dependency. In both cases, local times are infinitely divisible and we give a description of the corresponding Lévy measures in terms of excursion measures. These are hence analogues in the spectrally negative Lévy case of the first and second Ray-Knight theorems, originally stated for the Brownian motion.