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arXiv · 2004.10692

A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor opposite drift theorem

Thomas Gerard·Christophe Sabot·Xiaolin Zeng
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Abstract

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 a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, the distribution of these hitting times is related to a random potential that appears in the study of the vertex-reinforced jump process. The aim of this paper is to prove 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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Thomas Gerard, Christophe Sabot, Xiaolin Zeng. 2020-04-22. A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor opposite drift theorem. https://arxiv.org/abs/2004.10692

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