A variational approach to modeling slow processes in stochastic dynamical systems

TL;DR

This paper introduces a variational method to efficiently model slow dynamics in metastable stochastic systems by approximating eigenfunctions and eigenvalues, enabling better analysis and simulation of complex processes.

Contribution

It develops a variational principle based on a Rayleigh coefficient for approximating dominant eigenfunctions of Markov processes, facilitating efficient modeling of slow processes.

Findings

The approach estimates eigenfunctions from short simulations.

It provides a basis for adaptive algorithms for metastable systems.

Applicable to a wide range of stochastic processes.

Abstract

The slow processes of metastable stochastic dynamical systems are difficult to access by direct numerical simulation due the sampling problem. Here, we suggest an approach for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator. To this end, a variational principle is derived that is based on the maximization of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be estimated from statistical observables that can be obtained from short distributed simulations starting from different parts of state space. The approach forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem. Since any stochastic process with finite memory can be transformed into a Markov process, the approach is applicable to a wide range of processes relevant for modeling complex real-world phenomena.