Random Brownian Scaling and Some Absolute Continuity Relationships

Abstract

Consider (X _t ,t≥ 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 < 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

$$ X_t^{\left[ {a,b} \right]} \equiv \frac{1}{{\sqrt {b - a} }}{X_{a + t\left( {b - a} \right)}},t \leqslant 1, $$

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 ≤ 1), or the law of another interesting process (Y _t , t≤ 1), with a particularly simple density. In the following notes, I shall

• (a) present a list of such results; this is done in Section 1;

• (b) attempt to unify their proofs; this is partly done in Section 2;

• (c) 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;

• (d) develop some examples of applications to Bessel processes, in Section 4.