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<p>This paper studies the convergence rate of the Euler-Maruyama scheme for systems of interacting particles used to approximate solutions of nonlinear Fokker-Planck equations with singular interaction kernels, such as the Keller-Segel model. We derive explicit error estimates in the large-particle limit for two objects: the empirical measure of the interacting particle system and the density distribution of a single particle. Specifically, under certain assumptions on the interaction kernel and initial conditions, we show that the convergence rate of both objects towards solutions of the corresponding nonlinear Fokker-Planck equation depends polynomially on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( N \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation> (the number of particles) and on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( h \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>h</mi> </math></EquationSource> </InlineEquation> (the discretization step). The analysis shows that the scheme converges despite singularities in the drift term. To the best of our knowledge, there are no existing results in the literature of such kind for the singular kernels considered in this work.</p>

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Stochastic numerical approximation for nonlinear Fokker-Planck equations with singular kernels

  • 期刊论文

  • 发表日期：

    2026年7月27日

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  • N. Cazacu

摘要

This paper studies the convergence rate of the Euler-Maruyama scheme for systems of interacting particles used to approximate solutions of nonlinear Fokker-Planck equations with singular interaction kernels, such as the Keller-Segel model. We derive explicit error estimates in the large-particle limit for two objects: the empirical measure of the interacting particle system and the density distribution of a single particle. Specifically, under certain assumptions on the interaction kernel and initial conditions, we show that the convergence rate of both objects towards solutions of the corresponding nonlinear Fokker-Planck equation depends polynomially on \( N \) N (the number of particles) and on \( h \) h (the discretization step). The analysis shows that the scheme converges despite singularities in the drift term. To the best of our knowledge, there are no existing results in the literature of such kind for the singular kernels considered in this work.