Communications in Mathematical Physics

, Volume 336, Issue 3, pp 1259–1283 | Cite as

The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction

  • Scott HottovyEmail author
  • Austin McDaniel
  • Giovanni Volpe
  • Jan Wehr


We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein–Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.


Brownian Motion Stochastic Differential Equation Brownian Particle Stochastic Integral Lyapunov Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Scott Hottovy
    • 1
    • 2
    Email author
  • Austin McDaniel
    • 1
  • Giovanni Volpe
    • 3
    • 4
  • Jan Wehr
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA
  3. 3.Soft Matter Lab, Department of PhysicsBilkent UniversityAnkaraTurkey
  4. 4.UNAM-Institute of Material Science and NanotechnologyBilkent UniversityAnkaraTurkey

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