# Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations

## TL;DR

This paper introduces a Stratonovich-perturbed Langevin dynamics that maintains reversibility, accelerates convergence to equilibrium, and reduces asymptotic variance, enhancing sampling efficiency in high-dimensional probability measures.

## Contribution

It proposes a novel reversible Langevin dynamics with Stratonovich perturbations that improves sampling performance without the drawbacks of nonreversible methods.

## Key findings

- Achieves faster convergence to equilibrium.
- Reduces asymptotic variance in sampling.
- Demonstrates improved performance on a 2D warped Gaussian.

## Abstract

In this paper we propose a new approach for sampling from probability measures in, possibly, high dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in~[C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Ann. Appl. Probab. 1993] and studied in, e.g. [T. Lelievre, F. Nier, and G. A. Pavliotis J. Stat. Phys., 2013] and [A. B. Duncan, T. Leli\`evre, and G. A. Pavliotis J. Stat. Phys., 2016], while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution.