Quantitative homogenization of interacting particle systems

Abstract

For a class of interacting particle systems in continuous space, we show that finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. Our approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, we develop suitable modifications of the Caccioppoli and multiscale Poincaré inequalities, which are of independent interest.

Smoothness of the diffusion coefficients for particle systems in continuous space

“…Indeed, to show our main results, we only appeal to (2.5) as the definition of the limit diffusion matrix. This definition coincides with the more classical one based on full-space stationary correctors, as explained in [21,Appendix B].…”

“…For our purposes, it will be convenient to identify the bulk diffusion matrix as a limit of finite-volume approximations. In finite volume, there are in fact two natural approximations to the bulk diffusion matrix, which were introduced in [21] and are inspired by [8,10,31]. They are based on the following subadditive quantities: for every bounded domain U , p, q ∈ R d , and ρ 0 > 0, we define ν(U, p, ρ 0 ) ∶= inf…”

“…Informally, the functions in H 1 (U ) are those whose squared gradient have finite integral over U ; the functions in H 1 0 (U ) must in addition respond continuously to the exit from U or the entrance into U of a particle. One can check (see [21,Proposition 4.1] or Subsection 3.3 below) that there exist symmetric d-by-d matrices a(U, ρ 0 ), a * (U, ρ 0 ) that satisfy the bound (2.1) and such that, for every p, q ∈ R d , (2.4) ν(U, p, ρ 0 ) = 1 2 p ⋅ a(U, ρ 0 )p and ν * (U, q, ρ 0 ) = 1 2 q ⋅ a −1 * (U, ρ 0 )q.…”

“…It was shown in [21] that the sequence (a(◻ m , ρ 0 )) m∈N converges to the same limit, and moreover, that there exists an exponent α > 0 and a constant C < ∞ such that for every m ⩾ 1,…”

A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

“…Motivated by applications to homogenization of particle systems [5], Giunti, Gu, Mourrat, and Nitzschner recently addressed a similar problem, and proved the Gevrey regularity of λ → āλ in [6] (a variant of λ → Āλ , cf. Remark 2.2 below).…”