Asymptotic behaviour of some measure-valued diffusions

Thesis

Branching diffusions are introduced as a simple model of the growth of a population of rare mutant genes or of the spread of a small epidemic. Measure-valued critical branching diffusion processes are then constructed as the weak limits of the discrete particle branching diffusions when the scales of mass and lifetime are changed in an appropriate way. The distribution of such a measure-valued process is characterized as the unique solution to a martingale problem and this leads to an expression for its Laplace transform in terms of a deterministic dual process.

The dual process is given as the solution to a non-linear second order partial differential equation —the evolution equation of the measure-valued process. It is immediate that those measure-valued processes whose total mass is initially finite suffer extinction. The solution to the evolution equation is written in the form of a series whose coefficients are expressed in terms of the linear semigroup associated with the spatial diffusion part of the process. Estimates on these coefficients lead to a simple proof that if the spatial diffusion part of the measure-valued process is a recurrent (resp. transient) diffusion on R^d and the initial value of the process is a non-trivial invariant measure of this spatial diffusion then the process has no (resp. has a unique) non-trivial limiting distribution.

In order to overcome the tendency to extinction, a Poisson process of mass creation is introduced into the discrete particle branching diffusion. The resulting processes are rescaled in a way compatible with the previous rescaling to produce a measure-valued process with immigration. This process is also characterized via a deterministic dual, obtained by integrating the solution of the usual evolution equation with respect to time. The series solution to the evolution equation enables the moments to be calculated. It is shown that if the spatial diffusion is positive recurrent then the effect on the limit law of X(t)/t of immigrating at a constant rate is the same as that of conditioning the process without immigration on non-extinction. However, for a null recurrent diffusion it is shown that in the first case the limit is non-trivial (non-random) whereas in the second the limit is zero on any compact region.

Authors: Etheridge, A

• DOI: 10.5287/ora-xrjwrqjoj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English