Perpetual integrals via random time changes

Abstract

Let $(X_{t})_{t\geq0}$ be a $d$-dimensional Feller process with symbol $q$, and let $f:\mathbb{R}^{d}\to(0,\infty)$ be a continuous function. In this paper, we establish a growth condition in terms of $q$ and $f$ such that the perpetual integral \begin{equation*}\int_{0}^{\infty}f(X_{s})\,ds\end{equation*} is infinite almost surely. The result applies, in particular, if $(X_{t})_{t\geq0}$ is a Lévy process. The key idea is to approach perpetuals integrals via random time changes. As a by-product of the proof, a sufficient condition for the non-explosion of solutions to martingale problems is obtained. Moreover, we establish a condition which ensures that the random time change of a Feller process is a conservative $C_{b}$-Feller process.