# When are increment-stationary random point sets stationary?

2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
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.
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Journal articles

https://hal.inria.fr/hal-00863414
Contributor : Antoine Gloria <>
Submitted on : Wednesday, September 3, 2014 - 11:02:14 AM
Last modification on : Tuesday, July 3, 2018 - 11:41:25 AM
Long-term archiving on : Friday, April 14, 2017 - 3:59:54 PM

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### Citation

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⟩

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