Given a standard one-dimensional Brownian motion $B$, we define the hitting time as \begin{align*} \tau_a:=\inf\{t\geq0 \, : \, B_t = a\}, \end{align*} with $a \in \mathbb{R}$. From the continuity of the Brownian motion, we have that \begin{align*} B_{\tau_a}= a. \end{align*} Let $\tau_a^{-1}$ the right inverse of $\tau_a$. What can we say for $B_{\tau_a^{-1}}$ ?
We know that in law $\tau_a$ behaves as $1/2-$stable subordinator, then $\tau_a^{-1}$ in law is an inverse of an $1/2-$stable subordinator. I don't think we need it, since I have no independence between $B$ and $\tau_a$.
Do you have any suggestions?
Thank you very much.