## Abstract

We consider a mean-field system of path-dependent stochastic interacting diffusions in random media over a finite time window. The interaction term is given as a function of the empirical measure and is allowed to be nonlinear and path dependent. We prove that the sequence of empirical measures of the full trajectories satisfies a large deviation principle with explicit rate function. The minimizer of the rate function is characterized as the pathdependent McKean-Vlasov diffusion associated to the system. As corollary, we obtain a strong law of large numbers for the sequence of empirical measures. The proof is based on a decoupling technique by associating to the system a convenient family of product measures. To illustrate, we apply our results for the delayed stochastic Kuramoto model and for a SDE version of Galves-Löcherbach model.

Original language | English |
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Pages (from-to) | 665-695 |

Number of pages | 31 |

Journal | Annals of Applied Probability |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2022 |

### Bibliographical note

Publisher Copyright:© Institute of Mathematical Statistics, 2022.

### Funding

Funding. RB is supported by the Israel Science Foundation through grant 575/16 and by the German Israeli Foundation through grant I-1363-304.6/2016. AP was partially supported by Capes/PNPD fellowship 88882.315944/2019-01. GR is supported by a Capes/PNPD fellowship 888887.313738/2019-00. The authors thank IMPA for hospitality and financial support in the early stages of the work.

Funders | Funder number |
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German Israeli Foundation | 888887.313738/2019-00, 88882.315944/2019-01, I-1363-304.6/2016 |

Israel Science Foundation | 575/16 |

## Keywords

- McKean-Vlasov diffusions
- Path-dependent SDEs