2024/07/04 by José A. Cañizo, Cañizo, José A., Niccolò Tassi +1
Mathematics · Physics and Astronomy · #35B40 #35C10 #35R09 #45K05 #60F15 #60J76 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Probability (math.PR) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2407.03870
openalex publication_date 2024/07/04 · openalex created_date 2024/07/09 · openalex updated_date 2026/07/28
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.