### 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 language | English (US) |
---|---|

Pages (from-to) | 1259-1313 |

Number of pages | 55 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 69 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*69*(7), 1259-1313. https://doi.org/10.1002/cpa.21640

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

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 69, no. 7, pp. 1259-1313. https://doi.org/10.1002/cpa.21640

}

TY - JOUR

T1 - Large Deviations for Diffusions Interacting Through Their Ranks

AU - Dembo, Amir

AU - Shkolnikov, Mykhaylo

AU - Varadhan, Srinivasa

AU - Zeitouni, Ofer

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84963700612&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84963700612&partnerID=8YFLogxK

U2 - 10.1002/cpa.21640

DO - 10.1002/cpa.21640

M3 - Article

VL - 69

SP - 1259

EP - 1313

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 7

ER -