Mean Field Limit for Disordered Diffusions with Singular Interactions
نویسنده
چکیده
Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in R in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice Z, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter 0<α< d), we address the convergence as N →∞ of the empirical measure of the diffusions to the solution of a deterministic McKean–Vlasov equation and prove well-posedness of this equation, even in the degenerate case without noise. We provide also precise estimates of the speed of this convergence, in terms of an appropriate weighted Wasserstein distance, exhibiting in particular nontrivial fluctuations in the power-law case when d 2 ≤ α < d. Our framework covers the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators.
منابع مشابه
Spectral analysis of a family of symmetric, scale- invariant diffusions with singular coefficients and associated limit theorems
We discuss a family of time-reversible, scale-invariant diffusions with singular coefficients. In analogy with the standard Gaussian theory, a corresponding family of generalized characteristic functions provides a useful tool for proving limit theorems resulting in non-Gaussian, scale-invariant diffusions. We apply the generalized characteristic functions in combination with a martingale const...
متن کاملCugliandolo - Kurchan Equations for Dynamics of Spin - Glasses
We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of exp...
متن کاملMean-field Limit versus Small-noise Limit for Some Interacting Particle Systems
In the nonlinear diffusion framework, stochastic processes of McKean-Vlasov type play an important role. Such diffusions can be obtained by taking the hydrodymamic limit in a huge system of linear diffusions in interaction. In both cases, for the linear and the nonlinear processes, small-noise asymptotics have been emphasized by specific large deviation phenomenons. The natural question, theref...
متن کاملAn averaging principle for fast diffusions in domains separated by semi-permeable membranes
We prove an averaging principle which asserts convergence of diffusions on domains separated by semi-permeable membranes, when the speed of diffusion tends to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain’s intensities are proportional to membranes’ permeability and inve...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014