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

Nicoleta Cazacu

TL;DR

The paper develops quantitative convergence rates for an Euler–Maruyama discretization of a system of interacting particles that approximate nonlinear Fokker–Planck equations with singular kernels, including Keller–Segel-type interactions. By regularizing the kernel with a mollifier V^N and applying a drift cutoff F_A, the authors obtain uniform-in-N bounds and derive rates for both the mollified empirical measure and the density of a single particle toward the PDE solution, of the form O(N^{-v1} + N^{v2} h^{v3}) with explicit exponents depending on kernel regularity and dimension. The main results (Theorems 1 and 2) hinge on a semigroup/Itô expansion around Gaussian kernels and careful control of stochastic convolutions, yielding rates that hold in the large-N, small-h regime and hold for a broad class of singular kernels satisfying Krylov–Röckner-type conditions. These findings quantify the interplay between particle approximation and time discretization, providing guidance for choosing N and h in simulations of chemotaxis, electrostatics, and fluid-dynamic-inspired models where singular interactions arise.

Abstract

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 FokkerPlanck equation depends polynomially on N (the number of particles) and on 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.

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

TL;DR

The paper develops quantitative convergence rates for an Euler–Maruyama discretization of a system of interacting particles that approximate nonlinear Fokker–Planck equations with singular kernels, including Keller–Segel-type interactions. By regularizing the kernel with a mollifier V^N and applying a drift cutoff F_A, the authors obtain uniform-in-N bounds and derive rates for both the mollified empirical measure and the density of a single particle toward the PDE solution, of the form O(N^{-v1} + N^{v2} h^{v3}) with explicit exponents depending on kernel regularity and dimension. The main results (Theorems 1 and 2) hinge on a semigroup/Itô expansion around Gaussian kernels and careful control of stochastic convolutions, yielding rates that hold in the large-N, small-h regime and hold for a broad class of singular kernels satisfying Krylov–Röckner-type conditions. These findings quantify the interplay between particle approximation and time discretization, providing guidance for choosing N and h in simulations of chemotaxis, electrostatics, and fluid-dynamic-inspired models where singular interactions arise.

Abstract

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 FokkerPlanck equation depends polynomially on N (the number of particles) and on 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.

Paper Structure

This paper contains 9 sections, 5 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Notations and assumptions
  4. Main results
  5. Proofs of main results
  6. Preliminary results
  7. Proof of Theorem \ref{['thm.1']}
  8. Proof of Theorem \ref{['thm.2']}
  9. Examples

Key Result

Theorem 2.2

Under the assumptions assumption:A, assumption:AK and assumption:Aalpha, for any $\epsilon > 0$ and any $m \geq 1$, there exists a constant $C > 0$ such that for any $N \in \mathbb{N}^{\ast}$, for any $h>0$, where $\rho = \min(\alpha \zeta,\frac{1}{2}(1-\alpha(d + \chi_r)))$ and $\chi_r = \max(0, d(1-2/r))$.

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 3
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 1 more