# pr.probability – Conditional probability distribution of a Brownian particle surviving forever Answer

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## pr.probability – Conditional probability distribution of a Brownian particle surviving forever

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 $$\tau=\infty$$?

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