A trivariate law for certain processes related to perturbed Brownian motions

Abstract

D. Williams' path decomposition and Pitman's representation theorem for BES(3) are expressions of some deep relations between reflecting Brownian motion and the 3-dimensional Bessel process. In [Ph. Carmona et al., Stochastic Process. Appl. 7 (1999) 323–333], we presented an attempt to relate better reflecting Brownian motion and the 2-dimensional Bessel process, using space and time changes related to the Ray–Knight theorems on local times, in the manner of Jeulin [Lect. Notes Math., vol. 1118, Springer, Berlin, 1985] and Biane–Yor [Bull. Sci. Math. 2ème Sér. 111 (1987) 23–101]. Here, we characterize the law of a triplet linked to the perturbed Brownian motion which naturally arises in [Ph. Carmona et al., Stochastic Proc. Appl. 7 (1999) 323–333], and we point out its relations with Bessel processes of several dimensions. The results provide some new understanding of the generalizations of Lévy's arc sine law for perturbed Brownian motions previously obtained by the second author.