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In this paper, we study the mean field limit of interacting particles withmemory that are governed by a system of interacting non-Markovian Langevinequations. Under the assumption of quasi-Markovianity (i.e. that the memory inthe system can be described using a finite number of auxiliary processes), wepass to the mean field limit to obtain the corresponding McKean-Vlasov equationin an extended phase space. We obtain the fundamental solution (Green'sfunction) for this equation, for the case of a quadratic confining potentialand a quadratic (Curie-Weiss) interaction. Furthermore, for nonconvex confiningpotentials we characterize the stationary state(s) of the McKean-Vlasovequation, and we show that the bifurcation diagram of the stationary problem isindependent of the memory in the system. In addition, we show that theMcKean-Vlasov equation for the non-Markovian dynamics can be written in theGENERIC formalism and we study convergence to equilibrium and the Markovianasymptotic limit.

Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions