<p>The mean-field limit of interacting particle systems with singular interaction potentials under reflecting boundary conditions in non-convex domains is considered. To rigorously connect the microscopic particle system with its mesoscopic counterpart, we construct the empirical measure and establish stability by means of weighted exponential functionals. The main technical difficulties arise from the combination of non-convex geometry and singular particle interactions. To overcome the latter, we employ a smoothing (mollification) procedure of the interaction kernel, and then pass to the singular limit. For the boundary contribution, we exploit the exact shape of the boundary to keep the reflection terms under control. By combining moment bounds, stopping-time arguments, and mean-square error estimates for the empirical measure, we prove propagation of chaos and convergence to the reflected McKean-Vlasov equation.</p>

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Mean-Field Limit for Singular Potentials in Non-Convex Domains with Reflection

  • Yeqing Shan,
  • Guangying Lv

摘要

The mean-field limit of interacting particle systems with singular interaction potentials under reflecting boundary conditions in non-convex domains is considered. To rigorously connect the microscopic particle system with its mesoscopic counterpart, we construct the empirical measure and establish stability by means of weighted exponential functionals. The main technical difficulties arise from the combination of non-convex geometry and singular particle interactions. To overcome the latter, we employ a smoothing (mollification) procedure of the interaction kernel, and then pass to the singular limit. For the boundary contribution, we exploit the exact shape of the boundary to keep the reflection terms under control. By combining moment bounds, stopping-time arguments, and mean-square error estimates for the empirical measure, we prove propagation of chaos and convergence to the reflected McKean-Vlasov equation.