Exit probability of a Brownian particle. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:57:31Z http://mathoverflow.net/feeds/question/120512 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120512/exit-probability-of-a-brownian-particle Exit probability of a Brownian particle. Manas Patra 2013-02-01T12:40:59Z 2013-02-01T22:07:32Z <p>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,$\tau$]. </p> http://mathoverflow.net/questions/120512/exit-probability-of-a-brownian-particle/120526#120526 Answer by Yuri Bakhtin for Exit probability of a Brownian particle. Yuri Bakhtin 2013-02-01T15:49:18Z 2013-02-01T22:07:32Z <p>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.</p> <p>Also, see <a href="http://en.wikipedia.org/wiki/Doob" rel="nofollow">http://en.wikipedia.org/wiki/Doob</a>'s_martingale_inequality#Application:_Brownian_motion</p>