Abstract
We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system.
Funding Statement
E. Bayraktar was partially supported by the National Science Foundation under grant DMS-2106556 and by the Susan M. Smith chair.
Acknowledgments
We would like to thank the anonymous referees for their careful reading and many valuable suggestions on the paper.
Citation
Erhan Bayraktar. Suman Chakraborty. Ruoyu Wu. "Graphon mean field systems." Ann. Appl. Probab. 33 (5) 3587 - 3619, October 2023. https://doi.org/10.1214/22-AAP1901
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