When are increment-stationary random point sets stationary?

TL;DR

This paper investigates when increment-stationary random point sets are also stationary, providing conditions based on covariance decay and PDE theory, with different results depending on the spatial dimension.

Contribution

It characterizes the conditions under which increment-stationary point sets are stationary, extending understanding of their relationship through PDE and covariance decay analysis.

Findings

In dimensions d>2, increment-stationary sets are stationary up to a bounded translation.

In dimensions d=1 and d=2, such conditions do not exist.

Provides PDE-based criteria linking covariance decay to stationarity.

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.