Large Deviations for Diffusions Interacting Through Their Ranks

Amir Dembo, Mykhaylo Shkolnikov, Srinivasa Varadhan, Ofer Zeitouni

Research output: Contribution to journalArticle

Abstract

We prove a large deviations principle (LDP) for systems of diffusions (particles) interacting through their ranks when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the appropriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to a nondegenerate generalized porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of an LDP for interacting diffusions where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for tilted versions of such a generalized porous medium equation.

Original languageEnglish (US)
Pages (from-to)1259-1313
Number of pages55
JournalCommunications on Pure and Applied Mathematics
Volume69
Issue number7
DOIs
StatePublished - Jul 1 2016

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Interacting Diffusions
Large Deviations
Porous Medium Equation
Large Deviation Principle
Rate Function
Generalized Equation
Porous materials
McKean-Vlasov Equation
Vlasov equation
Cumulative distribution function
Unique Solution
Diffusion Coefficient
Distribution functions
Convection
Limiting
Regularity
Infinity
Tend
Interaction

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Large Deviations for Diffusions Interacting Through Their Ranks. / Dembo, Amir; Shkolnikov, Mykhaylo; Varadhan, Srinivasa; Zeitouni, Ofer.

In: Communications on Pure and Applied Mathematics, Vol. 69, No. 7, 01.07.2016, p. 1259-1313.

Research output: Contribution to journalArticle

Dembo, Amir ; Shkolnikov, Mykhaylo ; Varadhan, Srinivasa ; Zeitouni, Ofer. / Large Deviations for Diffusions Interacting Through Their Ranks. In: Communications on Pure and Applied Mathematics. 2016 ; Vol. 69, No. 7. pp. 1259-1313.
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