The Vervaat transform of Brownian bridges and Brownian motion

Abstract

For a continuous function $f\in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau (f)+t\mathrm{mod}1)+f(1)1_{\{t+\tau (f)\ge 1\}}-f(\tau (f))$, where $\tau (f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.

31 Pages, 4 figures. This paper (published by http://ejp.ejpecp.org/article/view/3744) is combined of two papers "On Vervaat transform of Brownian bridges and Brownian motion"(arXiv:1307.7952) by Jim Pitman and Wenpin Tang, and "Some points on Vervaat's transform of Brownian bridges and Brownian motion"(arXiv:1308.3759) by Titus Lupu

References in corpus (5)

• Poisson ensembles of loops of one-dimensional diffusions
• The Vervaat transform of Brownian bridges and Brownian motion
• The quantile transform of a simple walk
• Excursions Above the Minimum for Diffusions
• Shifting processes with cyclically exchangeable increments at random

Cited by in corpus (4)

• The Vervaat transform of Brownian bridges and Brownian motion
• The quantile transform of a simple walk
• Patterns in random walks and Brownian motion
• Some points on Vervaat's transform of Brownian bridges and Brownian motion