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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$\begingroup$ 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. $\endgroup$– Douglas ZareFeb 2, 2013 at 7:21
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$\begingroup$ 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. $\endgroup$– Manas PatraFeb 3, 2013 at 18:15
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This probability satisfies the heat equation on your interval with zero boundary condition and initial condition being identical 1. Solve it using Fourier series and separation of variables and you will obtain that the probability decays exponentially with exponent being the leading eigenvalue of the problem.
Also, see http://en.wikipedia.org/wiki/Doob's_martingale_inequality#Application:_Brownian_motion
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$\begingroup$ The probability density does satisfy the diffusion equation and the conditions imposed on it (nonnegativity, normalization and asymptotically 0) actually determines the solution. It is of course the normal density. The solution can be written as integration over paths. What I am looking for is a "good" estimate or lower bound. $\endgroup$ Feb 2, 2013 at 10:02
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$\begingroup$ @Manas Patra: The density is not the normal density. If it were, lower bounds would be very easy. $\endgroup$ Feb 2, 2013 at 13:00