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Conditional limit theorems for intermediately subcritical branching processes in random environment

V. I. Afanasyev, Ch. Böinghoff, G. Kersting, V. A. Vatutin (2014)

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

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment...

Connectivity bounds for the vacant set of random interlacements

Vladas Sidoravicius, Alain-Sol Sznitman (2010)

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

The model of random interlacements on ℤd, d≥3, was recently introduced in [Vacant set of random interlacements and percolation. Available at http://www.math.ethz.ch/u/sznitman/preprints]. A non-negative parameter u parametrizes the density of random interlacements on ℤd. In the present note we investigate connectivity properties of the vacant set left by random interlacements at level u, in the non-percolative regime u>u∗, with u∗ the non-degenerate critical parameter for the percolation...

Continuity of stochastic convolutions

Zdzisław Brzeźniak, Szymon Peszat, Jerzy Zabczyk (2001)

Czechoslovak Mathematical Journal

Let B be a Brownian motion, and let 𝒞 p be the space of all continuous periodic functions f ℝ → ℝ with period 1. It is shown that the set of all f ∈ 𝒞 p such that the stochastic convolution X f , B ( t ) = ∫ 0 t f ( t - s ) d B ( s ) , t ∈ [ 0 , 1 ] does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.

Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree

David Croydon (2008)

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

In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n→∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks...

Counting walks in a quadrant: a unified approach via boundary value problems

Kilian Raschel (2012)

Journal of the European Mathematical Society

The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations...

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