Further proba-bilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one sided travelling-waves

At the heart of this article will be the study of a branching Brownian motion (BBM) with killing, where individual particles move as Brownian motions with drift -rho, perform dyadic branching at rate beta and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the 'one-sided' FKPP travelling-wave equation of speed -rho with solutions f : [0, infinity) -) -> [0, 1] satisfying f (0) = 1 and f (infinity) = 0 has a unique solution with a particular asymptotic when rho < root 2 beta, and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503-517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53-72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is root 2 beta - rho on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W;, that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed lambda is an element of (0, root 2 beta - rho). Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed lambda > root 2 beta - rho. This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299-314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W-lambda as the limiting Radon-Nikodym derivative when conditioning the right-most particle to travel at speed into the distant future. (c) 2005 Elsevier SAS. All rights reserved.