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On the Langevin equation with variable friction

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Abstract

We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski–Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.

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Authors and Affiliations

  1. Faculty of Education and Integrated Arts and Sciences, Waseda University, Nishi-Waseda, Shinjuku, Tokyo, 169-8050, Japan

    Hitoshi Ishii

  2. Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL, 60657, USA

    Panagiotis E. Souganidis

  3. Department of Mathematics, The University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI, 53706, USA

    Hung V. Tran

Authors
  1. Hitoshi Ishii
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  2. Panagiotis E. Souganidis
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  3. Hung V. Tran
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Corresponding author

Correspondence to Hitoshi Ishii.

Additional information

Communicated by N. Trudinger.

Hitoshi Ishii: Partially supported by the KAKENHI No. 26220702, and No. 16H03948, JSPS.

Panagiotis E. Souganidis: Partially supported by the National Science Foundation Grants DMS-1266383 and DMS-1600129 and the Office for Naval Research Grant N00014-17-1-2095.

Hung V. Tran: Partially supported by the National Science Foundation Grants DMS-1615944 and DMS-1664424. This work began while at The University of Chicago as Dickson Instructor.

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Ishii, H., Souganidis, P.E. & Tran, H.V. On the Langevin equation with variable friction. Calc. Var. 56, 161 (2017). https://doi.org/10.1007/s00526-017-1240-7

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  • DOI: https://doi.org/10.1007/s00526-017-1240-7

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