Perpetual Integrals for Lévy Processes

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.