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, 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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Article dans une revue
Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (30), pp.1-14. 〈10.1214/ECP.v19-3288〉
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Contributeur : Antoine Gloria <>
Soumis le : mercredi 3 septembre 2014 - 11:02:14
Dernière modification le : mardi 3 juillet 2018 - 11:41:25
Document(s) archivé(s) le : vendredi 14 avril 2017 - 15:59:54

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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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