# Perpetual integrals for Lévy processes

Leif Doering, Andreas E. Kyprianou

Research output: Contribution to journalArticlepeer-review

6 Citations (SciVal)

## Abstract

Given a Lévy process $$\xi$$, we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral $$\int _0^\infty f(\xi _s)\hbox {d}s$$, where $$f$$ is a positive locally integrable function. If $$\mu =\mathbb {E}[\xi _1]\in (0,\infty )$$ and $$\xi$$ has local times we prove the 0–1 law \begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned} with the exact characterization \begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned} The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.
Original language English 1192-1198 7 Journal of Theoretical Probability 29 3 7 Mar 2015 https://doi.org/10.1007/s10959-015-0607-y Published - Sep 2016

## Fingerprint

Dive into the research topics of 'Perpetual integrals for Lévy processes'. Together they form a unique fingerprint.