February 2024 The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions
Renaud Raquépas
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 431-462 (February 2024). DOI: 10.1214/22-AIHP1336


We investigate the behaviour of a family of entropy production functionals associated to stochastic differential equations of the form


where b is a globally Lipschitz nonconservative vector field keeping the system out of equilibrium, with emphasis on the large-time limit and then the vanishing-noise limit. Different members of the family correspond to different choices of boundary terms. Our analysis yields a law of large numbers and a local large deviation principle which does not depend on the choice of boundary terms and which exhibits a Gallavotti–Cohen symmetry. We use techniques from the theory of semigroups and from semiclassical analysis to reduce the description of the asymptotic behaviour of the functional to the study of the leading eigenvalue of a quadratic approximation of a deformation of the infinitesimal generator near critical points of V.

Nous étudions le comportement d’une famille de fonctionnelles de production d’entropie associée aux équations différentielles stochastiques de la forme


b est un champ vectoriel Lipschitzien non conservatif qui maintient le système hors équilibre, en mettant l’accent sur la limite en temps long et la limite du bruit disparaissant, dans cet ordre. Les différents membres de la famille correspondent à des choix différents de termes de bords. Notre analyse donne une loi des grands nombres et un principe local de grandes déviations qui ne dépend pas du choix des termes de bords et qui présente une symétrie de Gallavotti–Cohen. Nous utilisons des techniques issues de la théorie des semigroupes et de l’analyse semi-classique pour réduire la description du comportement asymptotique de la fonctionnelle à l’étude à celle de la valeur propre principale d’une approximation quadratique d’une déformation du générateur infinitésimal près des points critiques de V.

Funding Statement

The work presented in this article was done while the author was a student at McGill University (Dept. of Mathematics and Statistics) and at Univ. Grenoble Alpes (Institut Fourier). During this preriod, the research of the author was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Agence Nationale de la Recherche through the grant NonStops (ANR-17-CE40-0006).


The author wishes to thank Vojkan Jakšić and Armen Shirikyan for guidance through the early stages of this project, as well as Noé Cuneo, Alain Joye and Claude-Alain Pillet for comments on earlier versions of this work.


Download Citation

Renaud Raquépas. "The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 431 - 462, February 2024. https://doi.org/10.1214/22-AIHP1336


Received: 21 June 2021; Revised: 11 July 2022; Accepted: 20 October 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718387
Digital Object Identifier: 10.1214/22-AIHP1336

Primary: 82C31 , 82C35
Secondary: 47D08 , 60H10

Keywords: Feynman–Kac semigroup , large deviations , Leading eigenvalue , semiclassical limit , Time reversal

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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