Mean-Field Limits for Stochastic Interacting Particles on Digraph Measures
Abstract: Many natural phenomena are effectively described by interacting particle systems, which can be modeled using either deterministic or stochastic differential equations (SDEs). In this study, we specifically investigate particle systems modeled by SDEs, wherein the mean field limit converges to a Vlasov-Fokker-Planck-type equation. Departing from conventional approaches in stochastic analysis, we explore the network connectivity between particles using diagraph measures (DGMs). DGMs are one possible tool to capture sparse, intermediate and dense network/graph interactions in the mean-field thereby going beyond more classical approaches such as graphons. Since the main goal is to capture large classes of mean-field limits, we set up our approach using measure-theoretic arguments and combine them with suitable moment estimates to ensure approximation results for the mean-field.
- N. Ayi and N. Pouradier Duteil. Mean-field and graph limits for collective dynamics models with time-varying weights. Journal of Differential Equations, 299:65–110, 10 2021.
- A. Backhausz and B. Szegedy. Action convergence of operators and graphs. Canad. J. Math., 74(1):72–121, 2022.
- Graphon mean field systems. The Annals of Applied Probability, 33(5):3587–3619, 2023.
- Weakly interacting oscillators on dense random graphs. Journal of Applied Probability, page 1–24, 2023.
- W. Braun and K. Hepp. The Vlasov dynamics and its fluctuation in the 1/n1𝑛1/n1 / italic_n limit of interacting particles. Communications in Mathematical Physics, 56:101–113, 06 1977.
- A new approach to the mean-field limit of Vlasov-Fokker-Planck equations. arXiv 2203.15747, 2023.
- L.-P. Chaintron and A. Diez. Propagation of chaos: a review of models, methods and applications. I. models and methods. Kinetic and Related Models, 15(6):895–1015, 2022.
- L.-P. Chaintron and A. Diez. Propagation of chaos: A review of models, methods and applications.ii. applications. Kinetic and Related Models, 15(6):1017–1173, 2022.
- H. Chiba and G. Medvedev. The mean field analysis of the kuramoto model on graphs i. the mean field equation and transition point formulas. Discrete and Continuous Dynamical Systems - A, 39, 12 2016.
- R.L. Dobrushin. Vlasov equations. Functional Analysis and Its Applications, 13(2):115–123, Apr 1979.
- Lawrence Evans. An introduction to stochastic differential equations. AMS, 2012.
- On choosing and bounding probability metrics. International Statistical Review, 70(3):419–435, 2002.
- M. Gkogkas and C. Kuehn. Graphop mean-field limits for kuramoto-type models. SIAM Journal on Applied Dynamical Systems, 21:248–283, 02 2022.
- F. Golse. On the Dynamics of Large Particle Systems in the Mean Field Limit, pages 1–144. 01 2016.
- Mean-field limit of non-exchangeable systems. arXiv preprint, arXiv:2112.15406., 2021.
- P.-E. Jabin and Z. Wang. Mean field limit and propagation of chaos for Vlasov systems with bounded forces. Journal of Functional Analysis, 271(12), 2016.
- C. Kuehn and C. Xu. Vlasov equations on digraph measures. Journal of Differential Equations, 339:261–349, 2022.
- G. Medvedev. The nonlinear heat equation on dense graphs and graph limits. SIAM Journal on Mathematical Analysis, 46(4):2743–2766, 2014.
- P. Mörters and Y. Peres. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2010.
- H. Neunzert. An Introduction to the nonlinear Boltzmann-Vlasov Equation, volume 1048, pages 60–110. 11 2006.
- T. Paul and E. Trélat. From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits. arXiv preprint, arXiv:2209.08832, 2022.
- A.S. Sznitman. Topics in propagation of chaos. In P.-L. Hennequin, editor, Ecole d’Eté de Probabilités de Saint-Flour XIX - 1989, volume 1464 of Lecture Notes in Mathematics, pages 165–251. Springer, 1991.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.