Electronic Communications in Probability

When are increment-stationary random point sets stationary?

Antoine Gloria

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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.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 30, 14 pp.

Accepted: 18 May 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 39A70: Difference operators [See also 47B39] 60H25: Random operators and equations [See also 47B80]

random geometry random point sets thermodynamic limit stochastic homogenization

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Gloria, Antoine. When are increment-stationary random point sets stationary?. Electron. Commun. Probab. 19 (2014), paper no. 30, 14 pp. doi:10.1214/ECP.v19-3288. https://projecteuclid.org/euclid.ecp/1465316732

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