Estimating the Most Probable Transition Time for Stochastic Dynamical Systems

TL;DR

This paper investigates the most probable transition time between metastable states in stochastic dynamical systems by analyzing the probability of the system's paths staying near transition trajectories, providing bounds and illustrative examples.

Contribution

It introduces a new approach focusing on the maximum probability of staying near transition paths, deriving bounds for transition times, and connecting with existing research.

Findings

Established exponential decay lower bound for transition probability

Derived power law decay upper bound for transition probability

Provided bounds for the most probable transition time in simple systems

Abstract

This work is devoted to the investigation of the most probable transition time between metastable states for stochastic dynamical systems. Such a system is modeled by a stochastic differential equation with non-vanishing Brownian noise, and is restricted in a domain with absorbing boundary. Instead of minimizing the Onsager-Machlup action functional, we examine the maximum probability that the solution process of the system stays in a neighborhood (or a tube) of a transition path, in order to characterize the most probable transition path. We first establish the exponential decay lower bound and a power law decay upper bound for the maximum of this probability. Based on these estimates, we further derive the lower and upper bounds for the most probable transition time, under suitable conditions. Finally, we illustrate our results in simple stochastic dynamical systems, and highlight the relation with some relevant works.