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<p>In this paper, the authors focus on the effective approximation (<i>N</i> →∞ then <i>ε</i> → 0) of stochastic interacting particle systems with fast regime-switching networks on digraph measures (DGMs). DGMs provide a robust approach to capturing sparse, intermediate, and dense network or graph interactions in the mean field, extending beyond traditional methods like graphons. The model can be used to simulate a vehicle’s trajectory under different traffic signal states. The main goals are to derive the simplified system (9) as <i>ε</i> → 0 and to capture a class of mean-field limits under the assumption that the switching process tends to a stationary state as time evolutions. Using the martingale method and validating the continuity of the underlying graph heterogeneity, the authors establish the convergence in law of (1) to a probability measure <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\overline \mu}_{t}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mover> <mi>μ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, which satisfies semi-linear Vlasov-Fokker-Planck equation.</p>

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Mean-Field Stochastic Interacting Particles with Fast Regime-Switching Networks on Digraph Measures

  • 期刊论文

  • 发表日期：

    2026年1月16日

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  • Haihan Yang,
  • Yan Lv,
  • Wei Wang

摘要

In this paper, the authors focus on the effective approximation (N →∞ then ε → 0) of stochastic interacting particle systems with fast regime-switching networks on digraph measures (DGMs). DGMs provide a robust approach to capturing sparse, intermediate, and dense network or graph interactions in the mean field, extending beyond traditional methods like graphons. The model can be used to simulate a vehicle’s trajectory under different traffic signal states. The main goals are to derive the simplified system (9) as ε → 0 and to capture a class of mean-field limits under the assumption that the switching process tends to a stationary state as time evolutions. Using the martingale method and validating the continuity of the underlying graph heterogeneity, the authors establish the convergence in law of (1) to a probability measure \({\overline \mu}_{t}\) μ ¯ t , which satisfies semi-linear Vlasov-Fokker-Planck equation.