Local times on rays for a class of planar Lévy processes

Abstract

LetX, Y be independent Lévy processes on the real line. AssumeX andY admit Lebesgue measure as a reference measure, thatP⁰(X_t>0)=c for allt>0 (or the weaker conditionP⁰(X_t>0)=c ast→∞) and thatY_t has a local time at points. We investigate the distribution of the local timeL_t of (X, Y) on the positivex-axis. It turns out that, under the first hypothesis (which is in particular satisfied by planar Brownian motion), ifT is an independent exponential time, then the ratio ofL_T toℓ_T, the local time on the entirex axis, is (generalized) arc-sine and independent ofℓ_T, andL_T has a Gamma distribution. We obtain then expressions for the distribution ofL_t. In the case of Brownian motion, the formula involves parabolic cylinder functions. Under the weaker condition mentioned above, together with mild secondary hypotheses, we obtain an expression for the asymptotic distribution ofL_t for larget.