New applications of spine constructions for branching diffusions

The mathematical objects we would like to investigate in this proposal are branching diffusions. These processes form a fundamental corner stone of the theory of stochastic processes and model the spatial and temporal evolution of a population in the presence of a confounding reproductive material. Given the birth of a particle, it dies at a rate which is proportional to a function of its spatial position and as long as a particle is alive, it moves independently according to a pre-specified diffusion process. At the end of its life, the particle dies and is replaced by a random number of particles situated at the parent particle's final position. The distribution law of the number of descendants can be spatially varying.As branching diffusions are themselves built from diffusions, it comes as no surprise that there are very intimate relations between the two both probabilistically and analytically. The many connections between branching diffusions and diffusions provides mathematical mechanisms by which one may study and understand the more complicated behaviour of the former of these two classes of stochastic processes. A probabilistic technique which connects the one to the other is the so called `spine construction'. Roughly speaking, given a branching diffusion, via a change of probability measure one obtains a new and non-homogenous spatial branching process in which an individual line of decent, the spine, has some exceptional stochastic behaviour (the spine's motion, branching rate and offspring distribution are all modified) whilst the behaviour of other individuals remains unchanged. An understanding of the behaviour of the resulting spine under the appropriate change of measure can be used to infer conclusions concerning the stochastic growth and spread of the branching process under the original measure. This spine approach has proved to be both very powerful and intuitive. Also closely related to branching diffusions are superprocesses. Superprocess arise as a short life time and high density limit of branching diffusions. For these processes analogues of `spine decompositions' also exist. The current project proposes a conceptual study of branching diffusions in the light of modern `spine technology'; ultimately with a view to extensions in the theory of superdiffusions. This includes an ambitious attempt to apply a spine (and hence a conceptual probabilistic) analysis to fundamental problems of spatial strong laws of large numbers for a very general class of branching diffusions and spatial growth of branching diffusions in random media (taking the form of `Poisson obstacles').The proposal requests a grant to cover the travel and subsistence for Prof. Janos Englander and Prof. Goetz Kesting for visits to the Department of Mathematical Sciences at Bath University to last eight weeks and two weeks, respectively, during the summer of 2007 for the purpose of scientific interaction. During this period, Prof. Engander will also give a short introductory course on branching diffusions, superdiffusions, random media and applications. The course will be integrated into the graduate programme for mathematics in Bath and an honourarium will be provided for the preparation of material.