### Abstract

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order m^{ℓ} ^{/} ^{2} over compact time intervals for any ℓ∈ Z^{+}. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the m→ 0 limit, which result in order m^{1 / 2} approximations. Our results cover bounded forces, for which we prove convergence in L^{p} norms and unbounded forces, in which case we prove convergence in probability.

Original language | English (US) |
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Pages (from-to) | 1765-1811 |

Number of pages | 47 |

Journal | Annales Henri Poincare |

Volume | 21 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2020 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics

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## Cite this

*Annales Henri Poincare*,

*21*(6), 1765-1811. https://doi.org/10.1007/s00023-020-00910-8