Invariance of Brownian motion associated with past and future maxima
Yuu Hariya
TL;DR
The paper studies invariance in law of Brownian motion under a family of anticipative path transformations built from an exponential functional $A_t=\int_0^t e^{2B_s}\,ds$. It shows that for every fixed $t>0$ and all $c>0$, the transform $\mathcal{T}^{(c)}$ leaves the law of Brownian paths on $[0,t]$ unchanged, and it identifies two extremal limits $\mathcal{G}$ and $\mathcal{M}$ which also preserve Brownian law; the latter is linked to Pitman’s $2M-X$ theorem. The paper establishes structural properties of these transforms (involution, time-reversal commutativity, and preservation of Pitman transforms), proves a limiting connection to $\mathcal{G}$ and $\mathcal{M}$, and provides a Wiener-measure disintegration via a parametric family $\mathcal{M}_x$ tied to past and future maxima. These results deepen the understanding of anticipative path transformations and their connections to classical results in Brownian motion and maximization principles with potential ties to representation theory and integrable systems.
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.