Capacities and efficient computation of first-passage probabilities.

Jackson Loper,Guangyao Zhou,Stuart Geman

doi:10.1103/physreve.102.023304

Jackson Loper, Guangyao Zhou + Show 1 more

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https://doi.org/10.1103/physreve.102.023304

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Abstract

A reversible diffusion process is initialized at position x_{0} and run until it first hits any of several targets. What is the probability that it terminates at a particular target? We propose a computationally efficient approach for estimating this probability, focused on those situations in which it takes a long time to hit any target. In these cases, direct simulation of the hitting probabilities becomes prohibitively expensive. On the other hand, if the timescales are sufficiently long, then the system will essentially "forget" its initial condition before it encounters a target. In these cases the hitting probabilities can be accurately approximated using only local simulations around each target, obviating the need for direct simulations. In empirical tests, we find that these local estimates can be computed in the same time it would take to compute a single direct simulation, but that they achieve an accuracy that would require thousands of direct simulation runs.

Highlights

Reversible diffusions play a key role in a wide variety of physical systems
If a first-passage probability is nearly the same for every initial condition in a region M, it can be well-approximated in terms of certain integrals
In Theorem 1 we show that the error introduced by this approximation is small whenever hitting probabilities are sufficiently similar across different initial conditions inside a large region, M

Summary

Introduction

Reversible diffusions play a key role in a wide variety of physical systems. The the physical state of the system at time t is represented through a variable Xt , confined to a finite region and evolving according to a stochastic differential equation. In this paper we seek to estimate first-passage probabilities (sometimes called “splitting probabilities” [7] or “hitting probabilities”) of such diffusions: given an initial condition x0 and a collection of targets, what is the probability that the process first hits any particular target before hitting any of the other targets? For example, multiple exit locations from a region of the state space in a statistical mechanics problem or the establishment of new intramolecular bonds in a folding problem.

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