# Exit probability of a Brownian particle.

Perhaps the answer is common folklore among probabilits and stochasticians(!)? But I would like a good lower estimate for the probability that a particle undergoing brownian motion in 1 dimensions stays in a given interval (say around 0 when the initial value of $x$ coordinate is 0) throughout a given time interval [0,$\tau$].

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If there is no drift, the exact answer can be expressed as an integral which isn't too complicated using the reflection principle. The probability density can be written as an alternating sum. –  Douglas Zare Feb 2 '13 at 7:21
The formal solution is known. In fact, ir is quite clearly stated in Wiener's original papers from the 1920's. And I repeat, I am looking for a good estimate that is some analytic expression in terms of the given parameters. I dont want to go the numerical way, yet. –  Manas Patra Feb 3 '13 at 18:15