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Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel

Niccolò Tassi

TL;DR

This work analyzes a linear nonlocal Fokker–Planck equation with heavy-tailed kernels, introducing a scaling parameter \\varepsilon\\ and a fractional order \\;s\\. It establishes exponential convergence to a unique equilibrium with a rate independent of \\varepsilon\\ and \\;s\\ by leveraging Harris’s theorem and a uniform generalised central limit theorem for heavy tails, yielding time-asymptotic preserving limits as \\varepsilon \ o 0\\ and \\;s \ o 1\\. The authors derive explicit convergence rates for the singular limit and the classical diffusion limit, show uniform-in-time convergence to the limiting equations, and provide precise regularity results for the equilibria. The results have implications for numerical schemes and for understanding the stability of nonlocal-outcome limits in probabilistic and kinetic contexts, especially regarding robustness across parameter regimes. Overall, the paper demonstrates that the long-time behavior of heavy-tailed nonlocal diffusion with confining drift remains stable under key parameter transitions, substantiating uniform limiting transitions and offering quantitative error bounds.

Abstract

We investigate the asymptotic behaviour of solutions of a class of nonlocal Fokker--Planck equations defined by nonsingular, heavy-tailed convolution kernels and characterised by a scaling parameter $\e\in(0,1]$ and a fractional index $s\in(1/2,1)$. By employing a suitable version of the generalised central limit for heavy-tailed distributions and the use of Harris's theorem, we prove exponential convergence to the equilibrium with a rate that is independent of both $\e$ and $s$. This allows us to show uniform--in--time convergence for both $\e\to 0$ and $s\to1$ recovering the limiting equations.

Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel

TL;DR

This work analyzes a linear nonlocal Fokker–Planck equation with heavy-tailed kernels, introducing a scaling parameter \\varepsilon\\ and a fractional order \\;s\\. It establishes exponential convergence to a unique equilibrium with a rate independent of \\varepsilon\\ and \\;s\\ by leveraging Harris’s theorem and a uniform generalised central limit theorem for heavy tails, yielding time-asymptotic preserving limits as \\varepsilon \ o 0\\ and \\;s \ o 1\\. The authors derive explicit convergence rates for the singular limit and the classical diffusion limit, show uniform-in-time convergence to the limiting equations, and provide precise regularity results for the equilibria. The results have implications for numerical schemes and for understanding the stability of nonlocal-outcome limits in probabilistic and kinetic contexts, especially regarding robustness across parameter regimes. Overall, the paper demonstrates that the long-time behavior of heavy-tailed nonlocal diffusion with confining drift remains stable under key parameter transitions, substantiating uniform limiting transitions and offering quantitative error bounds.

Abstract

We investigate the asymptotic behaviour of solutions of a class of nonlocal Fokker--Planck equations defined by nonsingular, heavy-tailed convolution kernels and characterised by a scaling parameter and a fractional index . By employing a suitable version of the generalised central limit for heavy-tailed distributions and the use of Harris's theorem, we prove exponential convergence to the equilibrium with a rate that is independent of both and . This allows us to show uniform--in--time convergence for both and recovering the limiting equations.
Paper Structure (21 sections, 31 theorems, 168 equations, 1 figure)

This paper contains 21 sections, 31 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Previous literature and related results
  4. Plan of the paper
  5. Preliminaries
  6. Fractional Calculus
  7. Fractional Laplacian
  8. Fractional diffusion and Fractional Fokker-Planck
  9. Examples of kernels
  10. Well-Posedness of the Initial Value Problem
  11. Representation of the solutions
  12. Fourier.
  13. Wild Sums.
  14. Berry-Esseen Central Limit Theorem for heavy-tailed distributions
  15. Asymptotic behaviour and properties of the equilibrium
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $(J^s)$ be a family of kernels satisfying Hypothesis ass: sunif. For any $\varepsilon \in (0,1)$, $s\in(1/2,1)$ and $k \in (0, 1]$:

Figures (1)

  • Figure 1: Quantitative convergence in $L^1_k$. The diagram illustrates that limits are uniform and independent of the specific convergence path taken.

Theorems & Definitions (66)

  • Remark 1.1
  • Theorem 1.2: Long-time asymptotics
  • Remark 1.3
  • Theorem 1.4: Non-singular to singular limit $\varepsilon \to 0$
  • Theorem 1.5: Long--range to short--range limit
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 56 more