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,$\tau$].
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