Absorption of a Randomly Accelerated Particle: Gambler's Ruin in a Different Game

We consider a particle which is randomly accelerated by Gaussian white noise on the line 0<x<1, with absorbing boundaries at x=0,1. Denoting the initial position and velocity of the particle by x_0 and v_0 and solving a Fokker-Planck type equation, we derive the exact probabilities q_0(x_0,v_0), q_1(x_0,v_0) of absorption at x=0,1, respectively. The results are in excellent agreement with computer simulations.