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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.
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Dates et versions

hal-03519829 , version 1 (10-01-2022)

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  • HAL Id : hal-03519829 , version 1

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Thomas Gerard, Christophe Sabot, Xiaolin Zeng. A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor opposite drift theorem. 2022. ⟨hal-03519829⟩
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