The random acceleration process in bounded geometries

In the random acceleration process considered here, a point particle is accelerated by Gaussian white noise with zero mean. Early work on first-passage and first-return probabilities and some recent applications are reviewed. Motion on the half line x > 0 and on the finite line 0 < x < 1 is considered, with two types of boundary conditions, absorbing and inelastic. In colliding inelastically with the boundary, the kinetic energy of the particle decreases. Results for this boundary condition are of interest in connection with driven granular matter. Some applications to semiflexible polymers are discussed where the statistics is the same as for the randomly accelerated particle. Finally, an introduction to the extreme statistics of the random acceleration process is given.