Consider the drift Brownian motion $X\_t:=1+bt+W\_t$, where $(W\_t)\_{t\ge 0}$ is a Brownian motion starting at zero. Set $\tau:=\inf\{t\ge 0: X\_t=0\}$. Assume $b>0$, then $\mathbb P[\tau=\infty]>0$. What is the conditional law of $X\_{\infty}$ knowing $\tau=\infty$?



Conditional probability distribution of a Brownian particle surviving forever
By the law of Large numbers, $X\_t/t \to b$ almost surely as $t \to +\infty$, hence $X\_t \to +\infty$ almost surely as $t \to +\infty$. Therefore $X\_\infty = +\infty$ almost surely under $P$ and also under $P[\cdot|\tau = \infty]$.


