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Large-scale isoperimetry on locally compact groups and applications

Romain Tessera (2006/2007)

Séminaire de théorie spectrale et géométrie

We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first...

Laslett’s transform for the Boolean model in ℝ d

Rostislav Černý (2006)

Kybernetika

Consider a stationary Boolean model X with convex grains in ℝ d and let any exposed lower tangent point of X be shifted towards the hyperplane N 0 = { x ∈ ℝ d : x 1 = 0 } by the length of the part of the segment between the point and its projection onto the N 0 covered by X . The resulting point process in the halfspace (the Laslett’s transform of X ) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie [Cressie])...

Law of large numbers for superdiffusions : the non-ergodic case

János Engländer (2009)

Annales de l'I.H.P. Probabilités et statistiques

In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of the Watanabe–Biggins LLN for super-brownian motion.

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