Abstract

Results on the behaviour of the rightmost particle in the nth generation in the branching random walk are reviewed and the phenomenon of anomalous spreading speeds, noticed recently in related deterministic models, is considered. The relationship between such results and certain coupled reaction-diffusion equations is indicated.

AMS subject classification (MSC2010) 60J80

Introduction

I arrived at the University of Oxford in the autumn of 1973 for postgraduate study. My intention at that point was to work in Statistics. The first year of study was a mixture of taught courses and designated reading on three areas (Statistics, Probability, and Functional Analysis, in my case) in the ratio 2:1:1 and a dissertation on the main area. As part of the Probability component, I attended a graduate course that was an exposition, by its author, of the material in Hammersley (1974), which had grown out of his contribution to the discussion of John's invited paper on subadditive ergodic theory (Kingman, 1973). A key point of Hammersley's contribution was that the postulates used did not cover the time to the first birth in the nth generation in a Bellman–Harris process. Hammersley (1974) showed, among other things, that these quantities did indeed exhibit the anticipated limit behaviour in probability. I decided not to be examined on this course, which was I believe a wise decisin but I was intrigued by the material. That interest turned out to be critical a few months later.