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.