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Escape over a saddle by colored noise: Theory and numerics

Jiayao Shao (邵家瑶)*, Tobias Grafke†, and Robert S. MacKay‡

  • *Contact author: jiayao.shao@warwick.ac.uk
  • †Contact author: T.Grafke@warwick.ac.uk
  • ‡Contact author: R.S.MacKay@warwick.ac.uk
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Phys. Rev. E 114, 024210 – Published 12 August, 2026

DOI: https://doi.org/10.1103/w8dd-9cqg

Abstract

Stochastic dynamical systems allow modeling of transitions induced by random disturbances, in particular from an attracting equilibrium and crossing the stable manifold of a saddle. While the small-noise limit is well-described by the large deviation principle, existing computational methods often struggle with stochastic forcings any more complicated than nondegenerate Gaussian white noise, and with unbounded time intervals. The primary innovations of this work are extending the framework to cater for colored and degenerate forcing and unbounded time horizons—scenarios that are physically realistic but numerically challenging. We cater for degenerate noise by using the Hamiltonian optimal control method. We cater for a class of colored noises by using linear filters on white noise. We cater for infinite time horizon by introducing the method of division (MOD), a computational approach for approximating rare transition events, including their most likely paths and the exponential scaling laws of their transition rates. The effectiveness of MOD and the above approaches to colored noise, is demonstrated by illustration on two examples: an inverted double-well potential and a simplified roll-heave model for ship capsize.

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Phys. Rev. E 114, 024210– Published 12 August, 2026
  • Received 28 February 2026
  • Accepted 9 July 2026
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