Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

Abstract

In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in~\cite{DuncanPavliotis2016}. We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

References in corpus (15)

• A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
• A consensus-based model for global optimization and its mean-field limit
• Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion
• Variance Reduction using Nonreversible Langevin Samplers
• Parameter Estimation for Multiscale Diffusions
• Generalized dynamical density functional theory for classical fluids and the significance of inertia and hydrodynamic interactions
• The McKean-Vlasov Equation in Finite Volume
• Large Deviations and Importance Sampling for Systems of Slow-Fast Motion
• The overdamped limit of dynamic density functional theory: Rigorous results
• Mean-Field Treatment of the Many-Body Fokker-Planck Equation
• Dimensional reduction in nonlinear filtering: A homogenization approach
• Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction
• A complete mean-field theory for dynamics of binary recurrent neural networks
• Noise-induced transitions in rugged energy landscapes
• Brownian Motion in an N-scale periodic Potential

Cited by in corpus (17)

• Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus
• On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
• Optimal Feedback Law Recovery by Gradient-Augmented Sparse Polynomial Regression
• Response Theory and Phase Transitions for the Thermodynamic Limit of Interacting Identical Systems
• Structure preserving schemes for the continuum Kuramoto model: phase transitions
• Dynamics of the Desai-Zwanzig model in multi-well and random energy landscapes
• Well-balanced finite volume schemes for hydrodynamic equations with general free energy
• Dimension reduction of noisy interacting systems
• Barriers of the McKean--Vlasov energy via a mountain pass theorem in the space of probability measures
• Population Extinction on a Random Fitness Seascape
• Response Theory Identifies Reaction Coordinates and Explains Critical Phenomena in Noisy Interacting Systems
• A Seascape Origin of Richards Growth
• Model reduction of Brownian oscillators: quantification of errors and long-time behaviour
• An invariance principle for gradient flows in the space of probability measures
• The sharp, the flat and the shallow: Can weakly interacting agents learn to escape bad minima?
• Mean-field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods
• A non-linear kinetic model of self-propelled particles with multiple equilibria