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The joint law of the last zeros of Brownian motion and of its Lévy transform

Published online by Cambridge University Press:  01 June 2000

CATHERINE DONATI-MARTIN ,
ZHAN SHI  and
MARC YOR
CATHERINE DONATI-MARTIN
Affiliation:
Laboratoire de Statistique et Probabilités, Université Paul Sabatier Toulouse III, 118 route de Narbonne, F-31062 Toulouse Cedex 04, France
ZHAN SHI
Affiliation:
Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05, France
MARC YOR
Affiliation:
Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05, France
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Abstract

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The joint study of functionals of a Brownian motion $B$ and its Lévy transform $\beta= |B|-L$, where $L$ is the local time of $B$ at zero, is motivated by the conjectured ergodicity of the Lévy transform.

Here, we compute explicitly the covariance of the last zeros before time one of $B$ and $\beta$, which turns out to be strictly positive.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 20 , Issue 3 , June 2000 , pp. 709 - 725
Copyright
2000 Cambridge University Press