Geometrical optics of first-passage functionals of random acceleration

Random acceleration is a fundamental stochastic process encountered in many applications. In the one-dimensional version of the process a particle is randomly accelerated according to the Langevin equation x ̈(t ) =√{2 D }ξ (t ) , where x (t ) is the particle's coordinate, ξ (t ) is Gaussian white noise with zero mean, and D is the particle velocity diffusion constant. Here, we evaluate the A →0 tail of the distribution P_n(A |L ) of the functional I [x (t ) ] =∫0_Txⁿ(t ) d t =A , where T is the first-passage time of the particle from a specified point x =L to the origin, and n ≥0 . We employ the optimal fluctuation method akin to geometrical optics. Its crucial element is determination of the optimal path—the most probable realization of the random acceleration process x (t ) , conditioned on specified A , n , and L . The optimal path dominates the A →0 tail of P_n(A |L ) . We show that this tail has a universal essential singularity, P_n(A →0 |L ) ∼exp(-α/_nL^{3 n +2} D A³ ) , where α_n is an n -dependent number which we calculate analytically for n =0 , 1, and 2 and numerically for other n . For n =0 our result agrees with the asymptotic of the previously found first-passage time distribution.