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

TL;DR

This paper analytically investigates the short-time statistics of first-passage functionals for high-order stochastic processes, revealing the tail behavior of their distributions using the optimal fluctuation method.

Contribution

It provides explicit analytical expressions for the most likely trajectories and tail distributions of first-passage functionals for high-order Langevin processes.

Findings

Derived the tail distribution form with an essential singularity at zero.

Obtained explicit exponents for the tail distribution for arbitrary process order.

Analyzed the most probable realizations of first-passage processes for specific functionals.

Abstract

We consider high-order stochastic processes $x(t)$ described by the Langevin equation $\frac{{{d^m}x\left( t \right)}}{{d{t^m}}}= \sqrt{2D} \xi(t)$, where $\xi(t)$ 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 $A=\int_{0}^{T} \left[ x(t)\right] ^n dt$ 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 $A \to 0$, $P_{m,n}(A |L) \sim \exp\left(-\frac{\alpha_{m,n}L^{2mn-n+2}}{D A^{2m-1}} \right)$, where the explicit expressions for the exponents $\alpha_{m,0}$ and $\alpha_{m,1}$ for arbitrary $m$ are obtained.