The Most Likely Transition Path for a Class of Distribution-Dependent Stochastic Systems

TL;DR

This paper investigates the most likely transition paths between stable states in distribution-dependent stochastic systems, simplifying the computation by relating it to non-distribution-dependent systems and demonstrating the method with examples.

Contribution

It introduces a novel approach to compute transition paths in distribution-dependent systems by reducing the problem to a non-dependent case, facilitating practical calculations.

Findings

The action functional simplifies in the small noise regime.

The method enables efficient computation of transition paths.

Illustrative examples demonstrate the approach's effectiveness.

Abstract

Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, the action functional does not involve the solution of the skeleton equation which describes the unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.