Smoothness of the diffusion coefficients for particle systems in continuous space

Abstract

For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of the corrector.

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).…”

“…In the introduction of [6], the authors point out that their approach based on Poisson calculus could be used to prove the regularity of λ → Āλ . Besides that regularity was in fact already proved a few years ago in [3], the approach of [6] is mostly a specific reformulation of the general results and arguments of [4] (based on the original triad local approximation / cluster expansions / improved ℓ 1 − ℓ 2 estimates) using Poisson calculus. The only new ingredient is a clever use of the independence of Poisson processes, which we have summarized in Lemma 2.3 below.…”

“…The only new ingredient is a clever use of the independence of Poisson processes, which we have summarized in Lemma 2.3 below. Note that [3] is not cited in [6] (it was announced in [4]), and that [4] is mentioned in [6] without detail and at the same level as [10] (which only treats first-order expansion in a discrete iid setting).…”

“…The aim of this note is to show that, although it might not look so a priori, the approach of [6] is indeed mostly a reformulation of [4] using Poisson calculus, and that the new ingredient of [6] can be efficiently and directly combined with the original and more general formulation of [4] to yield the Gevrey regularity of the map λ → Āλ (as well as of λ → āλ , see below).…”

“…• The authors first introduce a sequence of local approximations of āλ that only depend on the restriction of P λ on bounded domains [6,Section 3]. This is in line with the massive approximations used in [4].…”