Journal of Statistical Physics

, Volume 163, Issue 3, pp 659–673 | Cite as

The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and Positive Friction

  • David P. Herzog
  • Scott HottovyEmail author
  • Giovanni Volpe


A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to three physically realizable examples where the coefficients defining the Langevin equation for these examples grow unboundedly either at a boundary, such as a wall, and/or at the point at infinity. This unboundedness violates the assumptions of previous limit theorems in the literature. The main result of this paper proves convergence for such examples.


Small-mass limit Smoluchowski–Kramers approximation  Locally Lipschitz coefficients 



DH and SH would like to acknowledge support from Drake University and Iowa State University. GV was partially funded by a Marie Curie Career Integration Grant (PCIG11GA-2012-321726) and a Distinguished Young Scientist award of the Turkish Academy of Sciences (TÜBA).


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • David P. Herzog
    • 1
  • Scott Hottovy
    • 2
    Email author
  • Giovanni Volpe
    • 3
    • 4
  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.Soft Matter Lab, Department of PhysicsBilkent UniversityAnkaraTurkey
  4. 4.UNAM-Institute of Material Science and NanotechnologyBilkent UniversityAnkaraTurkey

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