Abstract
In this paper we prove a large deviation principle (LDP) for the empirical measure of a general system of mean-field interacting diffusions with singular drift (as the number of particles tends to infinity) and show convergence to the associated McKean–Vlasov equation. Along the way, we prove an extended version of the Varadhan Integral Lemma for a discontinuous change of measure and subsequently a LDP for Gibbs and Gibbs-like measures with singular potentials.
Dans cet article, nous prouvons un principe des grandes déviations (PGD) pour la mesure empirique d’un système général de diffusions interagissant en champ moyen avec une dérive singulière (lorsque le nombre de particules tend vers l’infini) et montrons la convergence vers l’équation de McKean–Vlasov associée. En cours de route, nous prouvons une version étendue du lemme intégral de Varadhan pour un changement discontinu de mesure et par la suite un PGD pour les mesures de type Gibbs avec des potentiels singuliers.
Funding Statement
J.H. and O.T. acknowledges support from NWO Vidi grant 016.Vidi.189.102, “Dynamical-Variational Transport Costs and Application to Variational Evolutions”. Parts of this work were undertaken when M.M. was at Weierstrass Institute for Applied Analysis and Stochastics, Germany, at Technische Universität Berlin, Germany, at University of York, UK, and at Università degli Studi di Milano, Italy. M.M. acknowledges support from the ECMAth via the Matheon project SE17 “Stochastic methods for the analysis of lithium-ion batteries”, from the Royal Society via the Newton International Fellowship NF170448 “Stochastic Euler equations and the Kraichnan model”, from project PRIN 2015233N54_002 “Deterministic and stochastic evolution equations” from the Italian Ministry of Education, University and Research, and from the Hausdorff Research Institute for Mathematics in Bonn under the Junior Trimester Program “Randomness, PDEs and Nonlinear Fluctuations”.
Acknowledgements
The authors thank M. A. Peletier for proposing the problem and for fruitful discussions at the start of the work.
Citation
Jasper Hoeksema. Thomas Holding. Mario Maurelli. Oliver Tse. "Large deviations for singularly interacting diffusions." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 492 - 548, February 2024. https://doi.org/10.1214/22-AIHP1319
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