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Let $B_t$ be a Brownian motion for a given probability space and $T:=\inf \lbrace t\geq 0 : \vert B_t \vert = 1 \rbrace$.

Is the process at this time, $B_T$, independent of the hitting time $T$? If so, how can one show this?

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    $\begingroup$ What is your reason for asking this question? In what context did it arise? Also, reading mathoverflow.net/howtoask and implementing the suggestions from there may help attract answers... $\endgroup$ – David Roberts Dec 21 '11 at 23:28
  • $\begingroup$ $B_T$ is random variable taking value 1 and -1 with probability 1/2, it is independent from $T$, you can conditionate by replacing T, by T' the first time B gets to 1, then B_T is a constant so it is independent of T'. Then taking T'' first time to -1 tell the same as T=min(T',T'') you get the result. Regards, but not an MO question more suited for mathstackexchange. $\endgroup$ – The Bridge Dec 21 '11 at 23:50
  • $\begingroup$ @The Bridge: I didn't understand your argument. Where does it fail for starting point other than 0? $\endgroup$ – Ori Gurel-Gurevich Dec 22 '11 at 0:20
  • $\begingroup$ This question may be easy, but I don't think the downvote is justified. $\endgroup$ – Ori Gurel-Gurevich Dec 22 '11 at 0:23
  • $\begingroup$ @Ori: I haven't downvoted, but perhaps the reason for the downvote was that a baldly asked question, devoid of context (what has the OP already tried, why did they come to this problem) can seem like homework. It is, to be fair, a natural question; but I think the question would be better received if the OP acted along the lines suggested by David Roberts $\endgroup$ – Yemon Choi Dec 22 '11 at 2:27
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In a word, "symmetry". (I presume you mean to have $B_0=0$.) The law of such a Brownian motion is invariant under orthogonal transformations, and the stopping time $T$ is pointwise invariant under such transformations. Therefore the law of $B_T$ is likewise invariant... This argument is valid in all dimensions.

More interesting is L. Pitt's converse, asserting that if the exit time from a bounded domain $D$ (for a Brownian motion started at $0\in D$) is independent of the exit place, then $D$ is essentially a ball centered at $0$. See [Annals of Probability, vol. 17 (1989), pp. 1651–1657].

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This result holds less obviously for Brownian motion with constant drift, not just $0$ drift. It is critical that the starting point is centered on the interval and it fails otherwise.

Stern, F. An Independence in Brownian Motion with Constant Drift. The Annals of Probability, Vol. 5 (1977), 571-572.

This holds for biased random walks because reflecting the paths to one boundary point gives paths to the other boundary with a constant magnification of probability. Taking the limit shows that the same is true for Brownian motion with constant drift.

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