Effective Langevin equations for constrained stochastic processes

TL;DR

This paper introduces a new stochastic method for exactly generating constrained Brownian paths conditioned on start and end points, using a local stochastic differential equation, applicable to various constrained processes and force fields.

Contribution

A novel, efficient stochastic approach for exact constrained path generation using a local SDE, applicable to multiple Brownian and Ornstein-Uehlenbeck processes.

Findings

Method efficiently generates independent constrained paths.

Applicable to Brownian bridges, meanders, excursions, and Ornstein-Uehlenbeck processes.

Demonstrated on paths connecting minima in a double-well potential.

Abstract

We propose a novel stochastic method to exactly generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time $t_{f}$. These paths are weighted with a probability given by the overdamped Langevin dynamics. We show how these paths can be exactly generated by a local stochastic differential equation. The method is illustrated on the generation of Brownian bridges, Brownian meanders, Brownian excursions and constrained Ornstein-Uehlenbeck processes. In addition, we show how to solve this equation in the case of a general force acting on the particle. As an example, we show how to generate constrained path joining the two minima of a double-well. Our method allows to generate statistically independent paths, and is computationally very efficient.