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

Benedict Leimkuhler; Matthias Sachs

doi:10.1007/978-3-030-15096-9_8

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

Benedict Leimkuhler^*, Matthias Sachs

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

5 Citations (Scopus)

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^∞ 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.

Original language	English
Title of host publication	Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017
Editors	Giambattista Giacomin, Stefano Olla, Ellen Saada, Herbert Spohn, Gabriel Stoltz, Gabriel Stoltz
Publisher	Springer
Pages	282-330
Number of pages	49
ISBN (Print)	9783030150952
DOIs	https://doi.org/10.1007/978-3-030-15096-9_8
Publication status	Published - 2019
Event	International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017 - Paris, France Duration: 12 Jun 2017 → 16 Jun 2017

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	282
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017
Country/Territory	France
City	Paris
Period	12/06/17 → 16/06/17

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© Springer Nature Switzerland AG 2019.

Keywords

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

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-030-15096-9_8

Cite this

Leimkuhler, B., & Sachs, M. (2019). Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force. In G. Giacomin, S. Olla, E. Saada, H. Spohn, G. Stoltz, & G. Stoltz (Eds.), Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017 (pp. 282-330). (Springer Proceedings in Mathematics and Statistics; Vol. 282). Springer. https://doi.org/10.1007/978-3-030-15096-9_8

Leimkuhler, Benedict ; Sachs, Matthias. / Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force. Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017. editor / Giambattista Giacomin ; Stefano Olla ; Ellen Saada ; Herbert Spohn ; Gabriel Stoltz ; Gabriel Stoltz. Springer, 2019. pp. 282-330 (Springer Proceedings in Mathematics and Statistics).

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

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

author = "Benedict Leimkuhler and Matthias Sachs",

note = "Funding Information: 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{\textquoteright}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. Publisher Copyright: {\textcopyright} Springer Nature Switzerland AG 2019.; International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017 ; Conference date: 12-06-2017 Through 16-06-2017",

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Leimkuhler, B & Sachs, M 2019, Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force. in G Giacomin, S Olla, E Saada, H Spohn, G Stoltz & G Stoltz (eds), Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017. Springer Proceedings in Mathematics and Statistics, vol. 282, Springer, pp. 282-330, International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017, Paris, France, 12/06/17. https://doi.org/10.1007/978-3-030-15096-9_8

Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force. / Leimkuhler, Benedict; Sachs, Matthias.
Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017. ed. / Giambattista Giacomin; Stefano Olla; Ellen Saada; Herbert Spohn; Gabriel Stoltz; Gabriel Stoltz. Springer, 2019. p. 282-330 (Springer Proceedings in Mathematics and Statistics; Vol. 282).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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AU - Leimkuhler, Benedict

AU - Sachs, Matthias

N1 - Funding Information: 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. Publisher Copyright: © Springer Nature Switzerland AG 2019.

PY - 2019

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N2 - 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∞ 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.

AB - 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∞ 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.

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KW - Heat-bath

KW - Molecular dynamics

KW - Mori-Zwanzig formalism

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KW - Quasi-Markovian model

KW - Reduced model

KW - Sampling

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A2 - Saada, Ellen

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A2 - Stoltz, Gabriel

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Y2 - 12 June 2017 through 16 June 2017

ER -

Leimkuhler B, Sachs M. Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force. In Giacomin G, Olla S, Saada E, Spohn H, Stoltz G, Stoltz G, editors, Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017. Springer. 2019. p. 282-330. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-030-15096-9_8

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

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this