Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

Properties of perpetual integral functionals of Brownian motion with drift

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Résumé: In this paper we study integrability properties of the random variables $$I_\infty(f):=\int_0^\infty f(B^{(\mu)}_t)\, dt,$$ where $\{B^{(\mu)}_t:t\geq~0\}$ is a Brownian motion with drift $\mu>0$ and $f$ is a non-negative, Borel measurable function. In particular, we find conditions under which $I_\infty(f)$ (i) is finite a.s., (ii) has all the moments, (iii) has some exponential moments, (iv) has bounded potential.