## The Annals of Applied Probability

### Equilibrium large deviations for mean-field systems with translation invariance

Julien Reygner

#### Abstract

We consider particle systems with mean-field interactions whose distribution is invariant by translations. Under the assumption that the system seen from its centre of mass be reversible with respect to a Gibbs measure, we establish large deviation principles for its empirical measure at equilibrium. Our study covers the cases of McKean–Vlasov particle systems without external potential, and systems of rank-based interacting diffusions. Depending on the strength of the interaction, the large deviation principles are stated in the space of centered probability measures endowed with the Wasserstein topology of appropriate order, or in the orbit space of the action of translations on probability measures. An application to the study of atypical capital distribution is detailed.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 2922-2965.

Dates
Revised: December 2017
First available in Project Euclid: 28 August 2018

https://projecteuclid.org/euclid.aoap/1535443238

Digital Object Identifier
doi:10.1214/17-AAP1379

Mathematical Reviews number (MathSciNet)
MR3847977

Zentralblatt MATH identifier
06974769

#### Citation

Reygner, Julien. Equilibrium large deviations for mean-field systems with translation invariance. Ann. Appl. Probab. 28 (2018), no. 5, 2922--2965. doi:10.1214/17-AAP1379. https://projecteuclid.org/euclid.aoap/1535443238

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