Periodicity and longtime diffusion for mean field systems in ℝ^d

2021/07/06 by Éric Luçon, Luçon, Eric, Christophe Poquet +1
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #35K55 #35Q84 #37N25 #60K35 #82C31 #92B20 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2107.02473

openalex publication_date 2021/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study in this paper the longtime behavior of some large but finite populations of interacting stochastic differential equations whose (infinite population) limit Fokker-Planck PDE admits a stable periodic solution. We show that the empirical measure for the population of size N stays close to the periodic solution, but with a random dephasing at the timescale Nt that converges weakly to a Brownian motion with constant drift.

Citations

• Small noise and long time phase diffusion in stochastic limit cycle\n oscillators
• Long time dynamics for interacting oscillators on graphs

Related

• Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding
• Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model
• Existence of Geometric Ergodic Periodic Measures of Stochastic Differential Equations
• Existence, stability and regularity of periodic solutions for nonlinear Fokker-Planck equations
• A Law of Large Numbers for interacting diffusions via a mild formulation