The Annals of Probability

Large deviation properties of weakly interacting processes via weak convergence methods

Amarjit Budhiraja, Paul Dupuis, and Markus Fischer

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Abstract

We study large deviation properties of systems of weakly interacting particles modeled by Itô stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean–Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 74-102.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325604997

Digital Object Identifier
doi:10.1214/10-AOP616

Mathematical Reviews number (MathSciNet)
MR2917767

Zentralblatt MATH identifier
1242.60026

Subjects
Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60B10: Convergence of probability measures 60H10: Stochastic ordinary differential equations [See also 34F05] 34K50: Stochastic functional-differential equations [See also , 60Hxx] 93E20: Optimal stochastic control

Keywords
Large deviations interacting random processes McKean–Vlasov equation stochastic differential equation delay weak convergence martingale problem optimal stochastic control

Citation

Budhiraja, Amarjit; Dupuis, Paul; Fischer, Markus. Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Probab. 40 (2012), no. 1, 74--102. doi:10.1214/10-AOP616. https://projecteuclid.org/euclid.aop/1325604997


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