Short-time large deviation of constrained random acceleration process

We consider the motion of a randomly accelerated particle in one dimension according to the Langevin equation x[over ̈](t)=sqrt[2D]ξ(t), where x(t) is the particle's position, ξ(t) is Gaussian white noise with zero mean, and D is the particle velocity diffusion constant. By employing the optimal fluctuation method, we study the short-time distribution P(A=A) of the functionals, A=∫_{0}^{t_{f}}x^{n}(t)dt, along constrained trajectories for a given time duration t_{f}, where n is a positive integer. We consider two types of constraints: one called the total constraint, where the initial position and velocity and the final position and velocity are both fixed, and the other, called the partial constraint, where the initial position and velocity and the final position are fixed, and letting the final velocity be free. Via the variation of constrained action functionals, the resulting Euler-Lagrange equations are analytically solved for n=1 and 2, and the optimal path, that is, the most probable realization of the random acceleration process x(t), conditioned on specified A and n, are correspondingly obtained. For n≥3, a numerical scheme is proposed to find the optimal path. We show that, for n=1, P(A) is a Gaussian distribution with the variance proportional to Dt_{f}^{5}. For n≥2, P(A) exhibits the non-Gaussian feature. In the small-A limit, P(A) shows an essential singularity, -lnP(A)∼A^{-3}, and the optimal path localizes around the initial state over a long time window and then escapes to the final position sharply at a late time. For A much larger than its typical value, there are multiple optimal paths with the same A but with different actions (or probability densities). Among these degenerate paths, the one with the minimum action is dominant and the others are exponentially unlikely. All theoretical results are validated by simulating the effective Langevin equations governing the constrained random acceleration process.