The Use of Hamiltonian Mechanics in Systems Driven by Colored Noise

TL;DR

This paper demonstrates that Hamiltonian mechanics can be applied to analyze stochastic systems driven by colored noise, providing better insights and numerical methods for calculating escape times in bistable potentials.

Contribution

It extends Hamiltonian formalism to colored noise-driven systems, enabling improved analysis and numerical computation of optimal paths and escape times.

Findings

Hamiltonian form applies to colored noise systems.

Oscillatory and unstable paths require multiple shooting techniques.

Caustics and bifurcations are common in stochastic optimal paths.

Abstract

The evaluation of the path-integral representation for stochastic processes in the weak-noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics. We show that, even when the noise is colored, these may be put into a Hamiltonian form which leads to better insights and improved numerical treatments. We concentrate on solving Hamilton's equations over an infinite time interval, in order to determine the leading order contribution to the mean escape time for a bistable potential. The paths may be oscillatory and inherently unstable, in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy. We look at two systems in some detail: the underdamped Langevin equation driven by external exponentially correlated noise and the overdamped Langevin equation driven by external quasi-monochromatic noise. We deduce that caustics, focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes.