A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor opposite drift theorem
Résumé
A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time change to this result, Matsumoto and Yor showed in [24] a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. In [19], Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector T 0 of hitting times of 0, also had the distribution of independent 3dimensional Bessel bridges. Moreover, the distribution of T 0 is related to a random potential β that appears in the study of the Vertex Reinforced Jump Process. The aim of this paper is to show a multivariate version of the Matsumoto-Yor opposite drift theorem, by applying a Lamperti-type time change to the previous family of interacting Brownian motions. Difficulties arise since the time change progresses at different speeds on different coordinates.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)