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Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force

  • Benedict LeimkuhlerEmail author
  • Matthias SachsEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 282)

Abstract

We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted \(L^{\infty }\) spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force.

Keywords

Generalized Langevin equation Heat-bath Quasi-Markovian model Sampling Molecular dynamics Ergodicity Central limit theorem Non-equilibrium Mori-Zwanzig formalism Reduced model 

Notes

Acknowledgements

The authors wish to thank Greg Pavliotis (Imperial), Jonathan Mattingly (Duke) and Gabriel Stoltz (ENPC) for their generous assistance in providing comments at various stages of this project. In particular, the authors thank Jonathan Mattingly for pointing out the possibility of using Girsanov’s theorem in the proof of Lemma 7. Both authors acknowledge the support of the European Research Council (Rule Project, grant no. 320823). BJL further acknowledges the support of the EPSRC (grant no. EP/P006175/1) during the preparation of this article. The work of MS was supported by the National Science Foundation under grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute.

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Copyright information

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Authors and Affiliations

  1. 1.The School of Mathematics and the Maxwell Institute of Mathematical Sciences, James Clerk Maxwell BuildingUniversity of EdinburghEdinburghUK
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.The Statistical and Applied Mathematical Sciences Institute (SAMSI)DurhamUSA

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