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Introduction to dynamical large deviations of Markov processes

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  • Touchette, Hugo

Abstract

These notes give a summary of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, defined in the context of Langevin-type equations, which model equilibrium and nonequilibrium processes driven by external forces and noise sources. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This analogy is used to explain the differences that exist between the fluctuations of equilibrium and nonequilibrium processes. An example involving the Ornstein–Uhlenbeck process is worked out in detail to illustrate these methods. Exercises, at the end of the notes, also complement the theory.

Suggested Citation

  • Touchette, Hugo, 2018. "Introduction to dynamical large deviations of Markov processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 504(C), pages 5-19.
    • Handle: RePEc:eee:phsmap:v:504:y:2018:i:c:p:5-19
    DOI: 10.1016/j.physa.2017.10.046
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    References listed on IDEAS

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    1. Mallick, Kirone, 2015. "The exclusion process: A paradigm for non-equilibrium behaviour," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 418(C), pages 17-48.
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    Cited by:

    1. Goswami, Koushik, 2021. "Work fluctuations in a generalized Gaussian active bath," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    2. Stutzer, Michael, 2020. "Persistence of averages in financial Markov Switching models: A large deviations approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    3. Nikolaos Limnios & Anatoliy Swishchuk, 2020. "Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications," Mathematics, MDPI, vol. 8(6), pages 1-16, June.
    4. Miguel Aguilera & Masanao Igarashi & Hideaki Shimazaki, 2023. "Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model," Nature Communications, Nature, vol. 14(1), pages 1-13, December.
    5. David T. Limmer & Chloe Y. Gao & Anthony R. Poggioli, 2021. "A large deviation theory perspective on nanoscale transport phenomena," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(7), pages 1-16, July.

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