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

Abstract

Given $a,b\ge 0$ and $t>0$, let $ρ=\{ ρ_{s}\} _{0\le s\le t}$ be a three-dimensional Bessel bridge from $a$ to $b$ over $[0,t]$. In this paper, based on a conditional identity in law between Brownian bridges stemming from Pitman's theorem, we show in particular that the process given by \begin{align*} ρ_{s}+\Bigl| b-a+ \min _{0\le u\le s}ρ_{u}-\min _{s\le u\le t}ρ_{u} \Bigr| -\Bigl| \min _{0\le u\le s}ρ_{u}-\min _{s\le u\le t}ρ_{u} \Bigr| ,\quad 0\le s\le t, \end{align*} has the same law as the time reversal $\{ ρ_{t-s}\} _{0\le s\le t}$ of $ρ$. As an immediate application, letting $R=\{ R_{s}\} _{s\ge 0}$ be a three-dimensional Bessel process starting from $a$, we obtain the following time-reversal and time-inversion results on $R$: $\{ R_{t-s}\} _{0\le s\le t}$ is identical in law with the process given by \begin{align*} R_{s}+R_{t}-2\min _{s\le u\le t}R_{u},\quad 0\le s\le t, \end{align*} when $a=0$, and $\{ sR_{1/s}\} _{s>0}$ is identical in law with the process given by \begin{align*} R_{s}-2(1+s)\min _{0\le u\le s}\frac{R_{u}}{1+u}+a(1+s),\quad s>0, \end{align*} for every $a\ge 0$.