Integral representations of certain measures in the one-dimensional diffusions excursion theory

Abstract

In this note we present integral representations of the Itô excursion measure associated with a general one-dimensional diffusion X. These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, BES(3), plays a crucial rôle in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair (RBM, BES(3)) generalize to pair.X; X­; where X­ denotes the diffusion obtained from X by conditioning X not to hit 0. We illustrate the results for the pair (R_, R+) consisting of a recurrent Bessel process with dimension d_ = 2(1-α), α Î(0,1), and a transient Bessel process with dimension d+ = 2(1+α). Pair (RBM, BES(3)) is, clearly, obtained by choosing α =1/2: