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We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching brownian motion with mild obstacles spreads less quickly than ordinary branching brownian motion by giving an upper estimate on its speed. When the underlying...

Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment

Jonathon Peterson (2009)

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

We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP>0 and there exists an s∈(1, 2) such that the annealed law of n−1/s(Xn−nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {tk} and {tk'} depending on the environment only, such that a quenched...

Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder

M. D. Jara, C. Landim (2008)

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

For a sequence of i.i.d. random variables {ξx: x∈ℤ} bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at x (resp. x+1) jumps to x+1 (resp. x) at rate ξx. We examine a quenched non-equilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder {ξx: x∈ℤ}. We prove that the position of the tagged particle converges under diffusive scaling to a...

Quermass-interaction process with convex compact grains

Kateřina Helisová, Jakub Staněk (2016)

Applications of Mathematics

The paper concerns an extension of random disc Quermass-interaction process, i.e. the model of discs with mutual interactions, to the process of interacting objects of more general shapes. Based on the results for the random disc process and the process with polygonal grains, theoretical results for the generalized process are derived. Further, a simulation method, its advantages and the corresponding complications are described, and some examples are introduced. Finally, a short comparison to the...