Celebratio Mathematica — Yor — Complete Bibliography

Complete Bibliography

Works connected to Erwin Bolthausen

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M. Yor: “Random Brownian scaling and some absolute continuity relationships,” pp. 243–252 in Seminar on stochastic analysis, random fields and applications (Ascona, Switzerland, 7–12 June 1993). Edited by E. Bolthausen, M. Dozzi, and F. Russo. Progress in Probability 36. Birkhäuser (Basel), 1995. MR 1360280 Zbl 0827.60010 incollection

Consider \( (X_t \), \( t\geq 0) \) a one-dimensional, or \( d \)-dimensional Brownian motion, or \( d \)-dimensional Bessel process, starting from 0, and let \( a \) and \( b \) be two random times with \( a \lt b \), almost surely. One finds, in the literature, a number of examples of such random couples \( (a,b) \) such that the law of the process \[ X_t^{[a,b]}\equiv \frac{1}{\sqrt{b-a}}X_{a+t(b-a)},\qquad t\leq 1, \] which is the transform of \( X \) by random Brownian scaling on the interval \( [a,b] \), is absolutely continuous with respect to the law of \( (X_t \), \( t\leq 1) \), or the law of another interesting process \( (Y_t \), \( t\leq 1) \), with a particularly simple density. In the following notes, I shall

present a list of such results; this is done in Section 1;
attempt to unify their proofs; this is partly done in Section 2;
show how these absolute continuity results may be applied to give a deep insight into P. Lévy’s arc sine law for Brownian motion; this is done in Section 3;
develop some examples of applications to Bessel processes, in Section 4.