When are increment-stationary random point ...
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
Title :
When are increment-stationary random point sets stationary?
Author(s) :
Gloria, Antoine [Auteur]
Département de Mathématique [Bruxelles] [ULB]
Quantitative methods for stochastic models in physics [MEPHYSTO]
Département de Mathématique [Bruxelles] [ULB]
Quantitative methods for stochastic models in physics [MEPHYSTO]
Journal title :
Electronic Communications in Probability
Pages :
1-14
Publisher :
Institute of Mathematical Statistics (IMS)
Publication date :
2014-05-18
ISSN :
1083-589X
English keyword(s) :
random geometry
random point sets
thermodynamic limit
stochastic homogenization.
stochastic homogenization
random point sets
thermodynamic limit
stochastic homogenization.
stochastic homogenization
HAL domain(s) :
Mathématiques [math]/Analyse numérique [math.NA]
English abstract : [en]
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 ...
Show more >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.Show less >
Show more >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.Show less >
Language :
Anglais
Popular science :
Non
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