Branching random walks, stable point processes and regular variation

Branching Processes and Random Walks

Branching-stable point processes

The notion of stability can be generalised to point processes by defining the scaling operation in a randomised way: scaling a configuration by t corresponds to letting such a configuration evolve according to a Markov branching particle system for -log t time. We prove that these are the only stochastic operations satisfying basic associativity and distributivity properties and we thus introdu...

Viral Processes by Random Walks on Random Regular Graphs

We study the SIR epidemic model with infections carried by k particles making independent random walks on a random regular graph. Here we assume k ≤ n , where n is the number of vertices in the random graph and is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdős-Rényi random graphs on the ...

Branching random walks and contact processes on Galton-Watson trees

We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds...

Random Walks and Branching Processes in Correlated Gaussian Environment

We study persistence probabilities for random walks in correlated Gaussian random environment first studied by Oshanin, Rosso and Schehr [27]. From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population ...