Invariance of Brownian motion associated with past and future maxima

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.