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[over ̈](t)=sqrt[2D]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}^{T}x^{n}(t)dt=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_{n}L^{3n+2}/DA^{3}), 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.