Chapter 2: The Spine Construction and the Strong Law of Large Numbers for branching diffusions

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.