# Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such that for $t\leq\tau$, we have \begin{align*} W_t=B_{\tau-t}-B_\tau \end{align*}

I am not sure if this can be done? I know about strong markov property and splicing brownian motion. I tried to use them to show this but didn't work. I thought at least similar proof to those theorems should apply here, but I couldn't make it work.

I think not. Let $B_t = (B_t^1, B_t^2)$ be your Brownian motion, and let $\tau = \inf\{t \ge 0 : B_t^1 = 1\}$. By recurrence of 1-D Brownian motion, $\tau < \infty$ almost surely. If $W_t = B_{\tau - t} - B_\tau$ for $t \le \tau$ as in your question, then $W$ has the property that almost surely, there exists a random time $\tau > 0$ such that $W_s^1 \le 0$ for all $0 \le s \le \tau$. The probability of a Brownian motion having this property is 0, so $W$ is not a Brownian motion.