Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials

Abstract

First, weak solutions of generalized stochastic Hamiltonian systems (gsHs) are constructed via essential m-dissipativity of their generators on a suitable core. For a scaled gsHs we prove convergence of the corresponding semigroups and tightness of the weak solutions. This yields convergence in law of the scaled gsHs to a distorted Brownian motion. In particular, the results confirm the convergence of the Langevin dynamics in the overdamped regime to the overdamped Langevin equation. The proofs work for a large class of (singular) interaction potentials including, e.g. potentials of Lennard-Jones type.

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Acknowledgements

The second author thanks the department of Mathematics at the University of Kaiserslautern for financial support in the form of a fellowship.

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Correspondence to Andreas Nonnenmacher.

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Grothaus, M., Nonnenmacher, A. Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials. J. Evol. Equ. 20, 577–605 (2020). https://doi.org/10.1007/s00028-019-00530-8

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Keywords

  • Markov semigroups
  • Langevin equations
  • Overdamped limit
  • Distorted Brownian motion
  • Semigroup convergence on varying spaces

Mathematics Subject Classification

  • 47D07
  • 60B12
  • 82C31