Skip to Main content Skip to Navigation
Journal articles

When are increment-stationary random point sets stationary?

Antoine Gloria 1, 2 
2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe, ULB - Université libre de Bruxelles
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.
Document type :
Journal articles
Complete list of metadata
Contributor : Antoine Gloria Connect in order to contact the contributor
Submitted on : Wednesday, September 3, 2014 - 11:02:14 AM
Last modification on : Thursday, March 24, 2022 - 3:12:58 AM
Long-term archiving on: : Friday, April 14, 2017 - 3:59:54 PM


Files produced by the author(s)




Antoine Gloria. When are increment-stationary random point sets stationary?. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (30), pp.1-14. ⟨10.1214/ECP.v19-3288⟩. ⟨hal-00863414v3⟩



Record views


Files downloads