Invariance of Brownian motion associated with past and future maxima

Abstract

Let $B=\{B_t{\}}_{t\ge 0}$ be a one-dimensional standard Brownian motion. As an application of a recent result of ours on exponential functionals of Brownian motion, we show in this paper that, for every fixed $t>0$, the process given by \begin{align*} B_{s}-B_{t}-\Bigl| B_{t}+\max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u} \Bigr| +\Bigl| \max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u} \Bigr| ,\quad 0\le s\le t, \end{align*} is a Brownian motion. The path transformation that describes the above process is proven to be an involution, commute with time reversal, and preserve Pitman's transformation. A connection with Pitman's $2M-X$ theorem is also discussed.

arXiv comment22 pages. This is the version accepted for publication in the journal of Stochastics on May 20, 2025; minor modifications have been made from the previous version

References in corpus (5)

• Exponential functionals of Brownian motion, I: Probability laws at fixed time
• Exponential functionals of Brownian motion, II: Some related diffusion processes
• Dynamics of the box-ball system with random initial conditions via Pitman's transformation
• Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings
• Discrete integrable systems and Pitman's transformation

Cited by in corpus (1)

• Invariance of three-dimensional Bessel bridges in terms of time reversal