TY - JOUR

T1 - Langevin Equations in the Small-Mass Limit

T2 - Higher-Order Approximations

AU - Birrell, Jeremiah

AU - Wehr, Jan

N1 - Funding Information:
J.W. was partially supported by NSF grant DMS 1615045. J.B. would like to thank Giovanni Volpe for suggesting this problem. J.B and J.W. would like to warmly thank the reviewers for their careful reading and many helpful suggestions for improving the presentation of this work.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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 mℓ / 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 m1 / 2 approximations. Our results cover bounded forces, for which we prove convergence in Lp norms and unbounded forces, in which case we prove convergence in probability.

AB - 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 mℓ / 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 m1 / 2 approximations. Our results cover bounded forces, for which we prove convergence in Lp norms and unbounded forces, in which case we prove convergence in probability.

UR - http://www.scopus.com/inward/record.url?scp=85084650055&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85084650055&partnerID=8YFLogxK

U2 - 10.1007/s00023-020-00910-8

DO - 10.1007/s00023-020-00910-8

M3 - Article

AN - SCOPUS:85084650055

VL - 21

SP - 1765

EP - 1811

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 6

ER -