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Consider a system of homogeneous interacting diffusive particles labeled bythe nodes of a unimodular Galton-Watson (UGW) tree, where the state of eachnode evolves like a d-dimensional diffusion whose drift coefficient depends on(the histories of) its own state and the states of neighboring nodes, and whosediffusion coefficient depends only on (the history of) its own state. Undersuitable regularity assumptions on the coefficients, an autonomouscharacterization is obtained for the marginal distribution of the dynamics ofthe neighborhood of a typical node in terms of a certain local equation, whichis a new kind of SDE that is nonlinear in the sense of McKean. This equationdescribes a finite-dimensional non-Markovian stochastic process whose evolutionat any time depends not only on the structure and current state of theneighborhood, but also on the conditional law of the current state given thepast of the states of neighborhing nodes. Such marginal distributions are ofinterest because they arise as weak limits of both marginal distributions andempirical measures of interacting diffusions on many sequences of sparse randomgraphs, including the configuration model and Erdos-Renyi graphs whose averagedegrees converge to a finite non-zero limit. The results obtained complementclassical results in the mean-field regime, which characterize the limitingdynamics of homogeneous interacting diffusions on complete graphs, as thenumber of nodes goes to infinity, in terms of a corresponding nonlinear Markovprocess. However, in the sparse graph setting, the topology of the graphstrongly influences the dynamics, and the analysis requires a completelydifferent approach. The proofs of existence and uniqueness of the localequation rely on delicate new conditional independence and symmetry propertiesof particle trajectories on UGW trees, as well as judicious use of changes ofmeasure.

Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees