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Title: Marginal dynamics of interacting diffusions on unimodular Galton–Watson trees
Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton–Watson tree, where the state of each node evolves infinitesi- mally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on (the history of) its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal dis- tribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the states of neighborhing nodes until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös–Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the num- ber of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph strongly influences the dynamics, and the analysis requires a completely different approach. The proofs of existence and uniqueness of the local equation rely on delicate new conditional independence and symmetry properties of particle trajectories on unimodular Galton– Watson trees, as well as judicious use of changes of measure.  more » « less
Award ID(s):
1954351
NSF-PAR ID:
10478863
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Probability Theory and Related Fields
Volume:
187
Issue:
3-4
ISSN:
0178-8051
Page Range / eLocation ID:
817 to 884
Subject(s) / Keyword(s):
Interacting diffusions · Sparse graphs · Random graphs · Local weak convergence · Mean-field limits · Nonlinear Markov processes · Erdo ̋s–Rényi graphs · Configuration model · Unimodularity · Markov random fields
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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