A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation
José A. Cañizo, Niccolò Tassi
TL;DR
This work establishes a uniform-in-time, nonlocal-to-local convergence framework for a scaled nonlocal Fokker-Planck equation driven by a kernel J_ε. By employing Harris's theorem, the authors prove exponential convergence to a unique equilibrium F_ε with constants independent of ε, and they quantify a uniform asymptotic-preserving convergence to the classical Fokker-Planck equation as ε→0 under symmetry. A novel L^∞ Berry–Esseen-based positivity bound and a Wild-sum representation underpin the positivity and constructive estimates, while a rigorous nonlocal-to-local analysis yields O(ε^2) convergence rates in appropriate weighted norms. The paper also thoroughly analyzes the equilibrium's regularity, moments, cumulants, and tail behavior, revealing how these features improve as ε decreases and clarifying the tails relative to Gaussian behavior. Overall, the approach provides a robust framework for nonlocal-diffusion limits with precise, explicit constants and clear connections to probabilistic limit theorems, offering a template for asymptotic-preserving analyses in nonlocal PDEs.
Abstract
We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which does not degenerate as we approach the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time-hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study.