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

This paper establishes explicit polynomial convergence rates for the Euler-Maruyama scheme applied to interacting particle systems approximating nonlinear Fokker-Planck equations with singular kernels, such as the Keller-Segel model, by deriving error estimates for both the empirical measure and single-particle density in the large-particle limit.

Original authors: Nicoleta Cazacu

Published 2026-08-07

📖 3 min read🧠 Deep dive

CC0 1.0

Original authors: Nicoleta Cazacu

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a bustling city where millions of people are moving around, each influenced by the crowd around them. If you wanted to predict where everyone will be in an hour, you couldn't track every single person individually; it would take too long and require too much computer power. Instead, you might use a clever trick: you simulate a smaller group of "representative" people who bump into each other, and you watch how their collective behavior spreads out to mimic the whole city. This is the heart of a field called stochastic particle methods. In science, this approach is used to solve complex math problems called Fokker-Planck equations, which describe how things like heat, chemicals, or even animals spread and interact over time.

However, things get tricky when the "rules of the road" for these particles are messy. Sometimes, the force one particle feels from another isn't smooth and gentle; it's sharp, sudden, and can even blow up to infinity if two particles get too close. Think of it like a magnet that pulls infinitely hard the moment it touches another magnet. In math, these are called singular kernels. They appear in real-world models for everything from how bacteria swarm to find food (chemotaxis) to how stars pull on each other via gravity. The big challenge for scientists has been: "Can we still use our computer simulations to get accurate answers when these forces are this wild and jagged?"

This paper tackles that exact question. The author, N. Cazacu, investigates a specific computer recipe called the Euler-Maruyama scheme. This is a step-by-step method for simulating how particles move over time. The problem is that when you try to use this recipe with "singular" (super-sharp) forces, the math usually breaks down or gives wildly wrong answers. The author proves that, surprisingly, you can make this recipe work, but they have to be very careful about how they smooth out the sharp edges and how many steps they take. They show that the error in the simulation shrinks at a predictable speed as you add more particles and make your time steps smaller. Specifically, they found that the accuracy depends on a delicate balance between the number of particles ( $N$ ) and the size of the time steps ( $h$ ). Even with these wild, singular forces, the simulation converges to the correct answer, provided you tune the parameters just right. They didn't just guess this; they provided rigorous mathematical proofs showing exactly how fast the error disappears, giving scientists a reliable roadmap for simulating these chaotic, singular systems.

Technical Summary: Stochastic Numerical Approximation for Nonlinear Fokker-Planck Equations with Singular Kernels

Problem Statement
This paper addresses the numerical approximation of solutions to nonlinear Fokker-Planck partial differential equations (PDEs) of the form:
$\partial_t u = \Delta u - \nabla \cdot (u K * u)$
where the interaction kernel $K$ is singular. Such equations arise in various physical and biological contexts, including the Keller-Segel chemotaxis model, Coulomb potentials in electrostatics/gravity, and the Biot-Savart law for 2D Navier-Stokes equations.

The primary challenge lies in the drift term $K * u$ , which involves a singular kernel. Standard numerical approaches, such as applying the Euler-Maruyama (EM) scheme directly to the associated McKean-Vlasov stochastic differential equation (SDE), face two difficulties:

Density Computation: The EM scheme requires evaluating the density of the solution at each time step, which is not directly available.
Singularity and Well-posedness: As the time step $h \to 0$ , the interaction kernel's singularities can compromise the well-posedness of the resulting particle system.

To overcome these, the author considers a system of $N$ interacting particles approximating the solution via an empirical measure. However, due to the singular drift, the system is regularized using a mollifier sequence $(V^N)$ and a smooth cut-off function $F_A$ to bound the drift, ensuring numerical stability and theoretical boundedness.

Methodology
The paper analyzes the convergence of the regularized, discretized particle system (Equation 1.4/2.2) towards the solution of the nonlinear Fokker-Planck equation. The methodology involves:

Regularization and Discretization:
- The singular kernel $K$ is convolved with a mollifier $V^N$ (scaling parameter $\alpha$ ).
- A cut-off function $F_A$ is applied to the drift to ensure uniform boundedness, consistent with the $L^\infty$ bound of the true drift $K * u_t$ .
- The system is discretized using the Euler-Maruyama scheme with time step $h$ .
Convergence Analysis Strategy:
The analysis decomposes the error between the mollified empirical measure $\tilde{\mu}^{N,h}_t$ (and the density of a single particle) and the true solution $u_t$ into several components:
- Initial Error: The difference between the initial mollified empirical measure and $u_0$ .
- Propagation of Chaos Error: The difference between the regularized particle system and the McKean-Vlasov SDE (non-discretized).
- Discretization Error: The error introduced by the EM scheme (time discretization).
- Stochastic Convolution: A term arising from the noise in the particle system.
Technical Tools:
- Semigroup Approach: The author utilizes the mild formulation of the Fokker-Planck equation involving the heat kernel $g_2(t, \cdot)$ .
- Itô's Formula: Applied to expand the densities of the particle system and the McKean-Vlasov solution around the heat kernel.
- Moment Bounds: The author establishes uniform bounds in $N$ for the mollified empirical measure in $L^p$ spaces.
- Gronwall's Lemma: Used to handle the integral inequalities arising from the error decomposition.
- Regularity Assumptions: The kernel $K$ is assumed to satisfy specific integrability conditions (Assumption AK) and a Hölder-type regularity condition on the convolution operator $K * f$ in terms of the $L^1 \cap L^r$ norm.

Key Contributions and Results
The paper derives explicit convergence rates for two objects in the large-particle limit ( $N \to \infty$ ) and the small-time-step limit ( $h \to 0$ ):

Convergence of the Mollified Empirical Measure (Theorem 2.5):
The paper establishes that the mollified empirical measure $\tilde{\mu}^{N,h}_t$ converges to the solution $u_t$ of the Fokker-Planck equation in the $L^1 \cap L^r$ norm. The convergence rate is given by:
$\mathbb{E}\left[ \sup_{s \in [0,t]} \| \tilde{\mu}^{N,h}_s - u_s \|^m_{L^1 \cap L^r} \right]^{1/m} \leq C \left( \mathbb{E}[\| \tilde{\mu}^{N,h}_0 - u_0 \|^m_{L^1 \cap L^r}]^{1/m} + N^{-\rho + \epsilon} + N^{v_2} h^{v_3} \right)$
where the exponents depend on the kernel properties and the mollification parameter $\alpha$ .
Convergence of the Particle Density (Theorem 2.6):
The paper provides a pointwise convergence rate for the density $u^{i,N,h}_t(x)$ of the $i$ -th particle towards the McKean-Vlasov density $u_t(x)$ . The bound includes a Gaussian weight term:
$|u^{i,N,h}_t(x) - u_t(x)| \leq C (\dots) \left( \int_{\mathbb{R}^d} g_{c/p}(t, x-z) u_0(dz) \right)^{1/p}$
This result quantifies the weak error for the EM scheme, even for test functions that can be Dirac masses.
Explicit Rates for Specific Kernels (Section 4):
The author applies their general results to specific cases:
- Bounded Lipschitz Kernels: Recovering rates of order $O(N^{-\frac{1}{d+2}} + h^{1/2})$ .
- Riesz Potentials and Coulomb Kernels: For $d \geq 2$ , the rate is $O(N^{-\frac{1}{2(d+1)}} + N^{\frac{d}{2(d+1)}} h^{1/2})$ .
- Krylov-Röckner Type Kernels: Similar rates are derived for kernels satisfying $K \in L^p$ with $p > d$ .

Significance and Claims
The paper claims that, to the best of the author's knowledge, these are the first results providing quantitative convergence rates for the Euler-Maruyama scheme applied to interacting particle systems with singular interaction kernels.

Handling Singularities: Unlike previous works that often require Lipschitz continuity of the drift in the Wasserstein metric (which fails for singular kernels), this work utilizes a specific regularity structure of the convolution operator $K * f$ in $L^1 \cap L^r$ spaces.
Discretization Analysis: While quantitative convergence for the non-discretized (continuous time) particle system was recently established by Olivera et al. [35], this paper extends the analysis to the discretized EM scheme, explicitly quantifying the trade-off between the number of particles $N$ and the time step $h$ .
Robustness: The method is shown to be robust for kernels that do not satisfy the standard Krylov-Röckner condition (e.g., highly singular Keller-Segel kernels where $\alpha \geq 2$ ), provided they satisfy the specific structural assumptions (AK) introduced in the paper.

The author emphasizes that the derived rates depend polynomially on $N$ and $h$ , and the analysis carefully tracks the dependence on the dimension $d$ and the singularity strength of the kernel. The work bridges the gap between theoretical propagation of chaos results and practical numerical implementation for singular interaction models.

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