Mean-field interacting systems with sequential coalescence at future ensemble averages

Abstract We introduce a new family of coalescent mean-field interacting particle systems by producing a pinning property that acts over a chosen sequence of multiple time segments. Throughout their evolution, these stochastic particles converge in time (i.e. get pinned) to their random ensemble average at the termination point of any one of the given time segments, only to burst back into life and repeat the underlying principle of convergence in each of the successive time segments, until they are fully exhausted. Although the architecture is represented by a system of piecewise stochastic differential equations, we prove that the conditions generating the pinning property enable every particle to preserve their continuity over their entire lifetime almost surely. As the number of particles in the system increases asymptotically, the system decouples into mutually independent diffusions, which, albeit displaying progressively uncorrelated behaviour, still close in on, and recouple at, a deterministic value at each termination point. Finally, we provide additional analytics including a universality statement for our framework, a study of what we call adjourned coalescent mean-field interacting particles, a set of results on commutativity of double limits, and a proposal of what we call covariance waves.