TY - JOUR

T1 - Geometrical optics of first-passage functionals of random acceleration

AU - Meerson, Baruch

N1 - Publisher Copyright:
© 2023 American Physical Society.

PY - 2023/6

Y1 - 2023/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85164024351&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.107.064122

DO - 10.1103/PhysRevE.107.064122

M3 - Article

C2 - 37464606

AN - SCOPUS:85164024351

SN - 2470-0045

VL - 107

JO - Physical Review E

JF - Physical Review E

IS - 6

M1 - 064122

ER -