Abstract
Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by Aldous and Bandyopadyay in discrete time. We here develop an analogue theory for recursive tree processes in continuous time. We illustrate the abstract theory on an example of a particle system with cooperative branching. This yields an interesting new example of a recursive tree process that is not endogenous.
Highlights
Let Ω and S be Polish spaces, let r be a finite measure on Ω with total mass |r| := r(Ω) > 0, and let γ : Ω × SN+ → S be measurable, where N+ := {1, 2, . . . }
As we have already mentioned, in Theorem 1.5 below, we prove that the mean-field limits of a large class of interacting particle systems are described by equations of the form (1.4)
We show that the endogeny of the Recursive Tree Process (RTP) corresponding to νlow and νupp follows from a general principle, discovered in [AB05], that says that Recursive Distributional Equation (RDE) that are defined by monotone maps always have a minimal and maximal solution with respect to the stochastic order, and that the
Summary
Let Ω and S be Polish spaces, let r be a finite measure on Ω with total mass |r| := r(Ω) > 0, and let γ : Ω × SN+ → S be measurable, where N+ := {1, 2, . . . }. As we have already mentioned, in Theorem 1.5 below, we prove that the mean-field limits of a large class of interacting particle systems are described by equations of the form (1.4) Aldous and Bandyopadhyay [AB05] define an RTP to be endogenous if the state at the root is a measurable function of the random maps attached to all vertices of the tree. They showed, in some precise sense (see Theorem 1.11 below), that endogeny is equivalent to stability of ν(n).
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