Measure-valued branching diffusions: immigrations, excursions and limit theorems

Limit Theorems for Some Branching Measure-valued Processes

We consider a particle system in continuous time, discrete population, with spatial motion and nonlocal branching. The offspring’s weights and their number may depend on the mother’s weight. Our setting captures, for instance, the processes indexed by a Galton-Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine this asymptotic behaviour. We also obtain a l...

Class of Limit Theorems for Singular Diffusions

Excursions and local limit theorems for Bessel - like random walks ∗

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x . We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form −δ...

Poisson representations of branching Markov and measure-valued branching processes

Representations of measure-valued processes in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle hits infinity. For each time t the joint distribution of l...

The Compact Support Property for Measure-valued Diffusions

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued diffusion processes corresponding to semi-linear equations of the form ut = Lu+ βu− αu p in R × (0,∞), p ∈ (1, 2]; u(x, 0) = f(x) in R; u(x, t) ≥ 0 in R × [0,∞). In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corres...