A relationship between three-dimensional Bessel processes in terms of Pitman's transformation

For every $x\ge 0$, let $R^x=\{R_t^x{\}}_{t\ge 0}$ be a three-dimensional Bessel process started at $x$. Given $x>0$, with the aid of Pitman’s theorem, a path transformation $T_x$ under which the process $T_x(R^y)$ is distributed as $R^x$ for any $y>0$ is constructed. In fact, we prove that, for every $x,y>0$, the two three-dimensional processes \begin{align*} \bigl( R^{x},T_{y}(R^{x}),$\mathcal{P}$(-R^{x})+x\bigr) && $\text{and}$ && \bigl( T_{x}(R^{y}),R^{y},$\mathcal{P}$(-R^{y})+y\bigr) \end{align*} are identical in law, where $\mathcal{P}$ is Pitman’s transformation. In the course of our exploration, it will be revealed that, for any $x>0$, the process $\mathcal{P}(-R^x)+x$ has the same law as the standard three-dimensional Bessel process $R^0$, which is then applied to derive a disintegration formula for the law of $R^x$ in terms of Pitman’s transformation. An application of the formula to three-dimensional Bessel bridges is also presented.