# Precursors to Rare Events in Stochastic Resonance

## TL;DR

This paper introduces a path-integral approach to predict rare transitions in stochastic resonance systems by identifying optimal paths and precursor fluctuations, enhancing the ability to forecast stochastic jumps.

## Contribution

It develops a novel method using path integrals and a comparison of Langevin and Hamiltonian dynamics to predict rare events in stochastic resonance.

## Key findings

- The method accurately predicts impending jumps in simulated systems.
- Optimal paths converge to deterministic minimizers near transitions.
- The framework is applicable to a broad class of stochastic resonance systems.

## Abstract

In stochastic resonance, a periodically forced Brownian particle in a double-well potential jumps between minima at rare increments, the prediction of which poses a major theoretical challenge. Here, we use a path-integral method to find a precursor to these transitions by determining the most probable (or "{optimal}") space-time path of a particle. We characterize the optimal path using a direct comparison principle between the Langevin and Hamiltonian dynamical descriptions, allowing us to express the jump condition in terms of the accumulation of noise around the stable periodic path. In consequence, as a system approaches a rare event these fluctuations approach one of the deterministic minimizers, thereby providing a precursor for predicting a stochastic transition. We demonstrate the method numerically, which allows us to determine whether a state is following a stable periodic path or will experience an incipient jump with a high probability. The vast range of systems that exhibit stochastic resonance behavior insures broad relevance of our framework, which allows one to extract precursor fluctuations from data.