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When are increment-stationary random point sets stationary?

par Gloria, Antoine
Référence Electronic communications in probability, 19
Publication Publié, 2014-05

Article révisé par les pairs

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Résumé :

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 incrementstationary 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.