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**Variance reduction for irreversible Langevin samplers and diffusion on graphs.**
*(English)*
Zbl 1347.60015

Summary: In recent papers, it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is, from several points of view, advantageous when compared to sampling from a time-reversible one. Adding an appropriate irreversible drift to the overdamped Langevin equation results in a larger large deviations rate function for the empirical measure of the process, a smaller variance for the long time average of observables of the process, as well as a larger spectral gap. In this work, we concentrate on irreversible Langevin samplers with a drift of increasing intensity. The asymptotic variance is monotonically decreasing with respect to the growth of the drift and we characterize its limiting behavior. For a Gibbs measure whose potential has one or more critical points, adding a large irreversible drift results in a decomposition of the process in a slow and fast component with fast motion along the level sets of the potential and slow motion in the orthogonal direction. This result helps understanding the variance reduction, which can be explained at the process level by the induced fast motion of the process along the level sets of the potential. Correspondingly, the limit of the asymptotic variance is the asymptotic variance of the limiting slow motion, which is a diffusion process on a graph.

### MSC:

60F05 | Central limit and other weak theorems |

60F10 | Large deviations |

60J60 | Diffusion processes |

60J25 | Continuous-time Markov processes on general state spaces |

65C05 | Monte Carlo methods |

82B80 | Numerical methods in equilibrium statistical mechanics (MSC2010) |