When are increment-stationary random point sets stationary?

Antoine Gloria

doi:10.1214/ecp.v19-3288

Abstract

In a recent work, Blanc, Le Bris, and Lions defined a notion of increment-stationarity for random point sets, which allowed them to prove the existence of a thermodynamic limit for two-body potential energies on such point sets (under the additional assumption of ergodicity), and to introduce a variant of stochastic homogenization for increment-stationary coefficients. Whereas stationary random point sets are increment-stationary, it is not clear a priori under which conditions increment-stationary random point sets are stationary. In the present contribution, we give a characterization of the equivalence of both notions of stationarity based on elementary PDE theory in the probability space. This allows us to give conditions on the decay of a covariance function associated with the random point set, which ensure that increment-stationary random point sets are stationary random point sets up to a random translation with bounded second moment in dimensions $d>2$. In dimensions $d=1$ and $d=2$, we show that such sufficient conditions cannot exist.

Highlights

In [2], Blanc, Le Bris and Lions addressed the issue of defining the thermodynamic limit of the energy of random sets l of particles
Let l : Ω → L(Rd) be an increment-stationary random point set for the group action {θk}k∈Zd
In terms of the random point set l, this turns into: For all i, j, k ∈ Z and almost every ω ∈ Ω, Xi+k(ω) − Xj+k(ω) = Xi(θkω) − Xj. The latter implies that the random point set l is increment-stationary with Yk(ω) = X0(θkω) − X0(ω)

Summary

Introduction

We say that l is increment-stationary if there exist a sequence of random vectors {Yk}k∈Zd of L2(Ω, Rd) and a measure-preserving group action {θk}k∈Zd such that l satisfies for almost all ω ∈ Ω and all k ∈ Zd, l(θkω) = l(ω) + Yk(ω) that is, l(θkω) is the translation of l(ω) by the vector Yk(ω). Let l : Ω → L(Rd) be an increment-stationary random point set for the group action {θk}k∈Zd .

Results

Conclusion