Spatial Branching in Random Environments and with Interaction

The following sections are included:

• Dedication
• Preface
• Contents

This book discusses some models involving spatial motion, branching, random environments and interactions between particles, in domains of the d-dimensional Euclidean space. These models are often easy to grasp intuitively and in fact, they dovetail very nicely with certain population models. Still, working with them requires some background in advanced probability and analysis. In this chapter, therefore, we will review the preliminaries

I take it for granted that the reader has a measure theoretical background and is familiar with some general concepts for stochastic processes. With regard to measure theory, I am writing with the expectation that the reader has undertaken, for example, a standard graduate level measure theory course at a US university. For stochastic processes, I assume that the reader has had a graduate level probability course and has been exposed, for instance, to the concept of finite dimensional distributions of a process, Kolmogorov's Consistency Theorem, and to the fundamental notions of martingales and Markov processes in continuous time.

We start with frequently used notation.

In this chapter we study a strictly dyadic branching diffusion Z corresponding to the operator Lu + β(u² − u) on D ⊆ ℝ^d (where β is as in (1.35)). Our main purpose is to demonstrate that, when λ_c ∈ (0,∞) and L + β − λ_c possesses certain ‘criticality properties,’ the random measures e^−λ_ctZt converge almost surely in the vague topology as t → ∞. As before, λ_c denotes the generalized principal eigenvalue for the operator L + β on D

The reason we are considering vague topology instead of the weak one, is that we are investigating the local behavior of the process. As it turns out, local and global behaviors are different in general.

As a major tool, the 'spine change of measure' is going to be introduced; we believe it is of interest in its own right.

In this chapter we provide examples which satisfy all the assumptions we had in the previous chapter, and thus, according to Theorem 2.2, obey the SLLN.

We begin with discussing the important particular case when the domain is bounded.

In this chapter our goal is to obtain a strong (a.s.) limit theorem for a spatial branching system, but with a new feature: the particles, besides moving and branching, now also interact with each other.

To this end, we introduce a branching Brownian motion (BBM) with 'attraction' or 'repulsion' between the particles.

Recall the problem of a single Brownian particle trying to ‘survive’ in a Poissonian system of traps (obstacles) from Section 1.12. In many models (for example those in population biology or nuclear physics), the particles also have an additional feature: branching. It seems therefore quite natural to ask whether an asymptotics similar to (1.29) can be obtained for branching processes. Analogously to the single particle case, one may want to study, for example, the probability that no reaction has occurred up to time t > 0 between the family generated by a single particle and the traps.

Thus, in this chapter we will study a branching Brownian motion on ℝ^d with branching rate β > 0, in a Poissonian field of traps.

In Section 1.12, we reviewed the problem of the survival asymptotics for a single Brownian particle among Poissonian obstacles — an important topic, thoroughly studied in the last few decades. As mentioned in the notes to the previous chapter, more recently a model of a spatial branching process in a random environment has been introduced; ‘hard’ obstacles were considered, and instantaneous killing of the branching process once any particle hits the trap configuration K. In the previous chapter we have seen how the model can result in some surprising phenomena when the trap intensity varies in space, and we also discussed some questions when killing is defined as the absorption of the last particle…

In the previous chapter we have studied a spatial branching model, where the underlying motion was a d-dimensional (d ≥ 1) Brownian motion, the particles performed dyadic branching, and the branching rate was affected by a random collection of reproduction suppressing sets; the obstacle configuration was given by the union of balls with fixed radius, where the centers of the balls formed a Poisson point process…

The following sections are included:

• Appendix A Path continuity for Brownian motion
• Appendix B Semilinear maximum principles
• Bibliography