Short-time large deviations of first-passage functionals for high-order stochastic processes

2024/09/27 by Tian, Lulu, Chen, Hanshuang, Li, Guofeng
#FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2409.18398

Abstract

We consider high-order stochastic processes x(t) described by the Langevin equation \fracd^mx( t )dt^m= √(2D) ξ(t), where ξ(t) is a delta-correlated Gaussian noise with zero mean, and D is the strength of noise. We focus on the short-time statistics of the first-passage functionals A=∫₀^T [ x(t)] ⁿ dt along the trajectories starting from x(0)=L and terminating whenever passing through the origin for the first-time at t=T. Using the optimal fluctuation method, we analytically obtain the most likely realizations of the first-passage processes for a given constraint A with n=0 and 1, corresponding to the first-passage time itself and the area swept by the first-passage trajectory, respectively. The tail of the distribution of A shows an essential singularity at A → 0, P_m,n(A |L) ∼ exp(-\fracα_m,nL^2mn-n+2D A^2m-1 ), where the explicit expressions for the exponents α_m,0 and α_m,1 for arbitrary m are obtained.

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