I have narrow escape problem here. But for the sake of completeness, it measures the time it takes for a Brownian particle, $\alpha$, escape through a narrow hole, $A$, on an otherwise reflecting surface, enclosing a volume $V$, provided that $a\ll V^{1/3}$, where $a$ is the largest dimension of $A$.
Assume that know there is a probability distribution $p(t)$ that gives the probability that $\alpha \in V$ at $t$. We learn that $\alpha \in V$ at $T$, but we don't know its exact location in $V$.
Can we assume that $p(t+T|(\alpha \in V)_{t=T})=p(t)$, without an explicit knowledge of $p(t)$, relying on the fact that Brownian motion itself is time independent?