## Abstract

In this paper, we study the mean field limit of weakly interacting particles with memory that are governed by a system of non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. that the memory in the system can be described using a finite number of auxiliary processes), we pass to the mean field limit to obtain the corresponding McKean-Vlasov equation in

an extended phase space. For the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction, we obtain the fundamental solution (Green's function). For nonconvex confining potentials, we characterize the stationary state(s) of the McKean-Vlasov equation, and we show that the bifurcation diagram of the stationary problem is independent of the memory in the system. In

addition, we show that the McKean-Vlasov equation for the non-Markovian dynamics can be written in the GENERIC formalism and we study convergence to equilibrium and the Markovian asymptotic limit.

an extended phase space. For the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction, we obtain the fundamental solution (Green's function). For nonconvex confining potentials, we characterize the stationary state(s) of the McKean-Vlasov equation, and we show that the bifurcation diagram of the stationary problem is independent of the memory in the system. In

addition, we show that the McKean-Vlasov equation for the non-Markovian dynamics can be written in the GENERIC formalism and we study convergence to equilibrium and the Markovian asymptotic limit.

Original language | English |
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Pages (from-to) | 2199 – 2230 |

Journal | Communications in Mathematical Sciences |

Volume | 16 |

Issue number | 8 |

DOIs | |

Publication status | Published - 18 Apr 2019 |

## Keywords

- mean field limits
- non-Markovian interacting particles
- convergence to equilibrium
- GENERIC
- asymptotic limit
- phase transitions