Geometrical optics of first-passage functionals of random acceleration

DOI10.1103/PHYSREVE.107.064122arXiv2302.04029MaRDI QIDQ6425828

Baruch Meerson

Publication date: 8 February 2023

Abstract: 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 $ddotx(t)=sqrt2Dxi(t)$, where $x(t)$ is the particle's coordinate, $xi(t)$ is Gaussian white noise with zero mean, and $D$ is the particle velocity diffusion constant. Here we evaluate the $Ao0$ tail of the distribution $P_n(A|L)$ of the functional $I[x(t)]=in{t}_0^T{x}^n(t)dt=A$, where $T$ is the first-passage time of the particle from a specified point $x=L$ to the origin, and $ngeq0$. 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 $Ao0$, $n$ and $L$. This realization dominates the probability distribution $P_n(A|L)$. We show that the $Ao0$ tail of this distribution has a universal essential singularity, $P_n(Ao0|L)simexpleft(-fracalph{a}_n{L}^{3n+2}D{A}^{3}ight)$, where $alph{a}_n$ 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.

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