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 âˆ’Ï�, perform dyadic branching at rate Î² 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 â€˜one-sidedâ€™ FKPP travelling-wave equation of speed âˆ’Ï� with solutions satisfying f(0)=1 and f(âˆž)=0 has a unique solution with a particular asymptotic when , 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) 503â€“517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. PoincarÃ© Probab. Statist. 40 (1) (2004) 53â€“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 on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, WÎ», 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 . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed . 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) 299â€“314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals WÎ» as the limiting Radonâ€“NikodÃ½m derivative when conditioning the right-most particle to travel at speed Î» into the distant future.
|Translated title of the contribution||Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one sided travelling-waves|
|Pages (from-to)||125 - 145|
|Number of pages||21|
|Journal||Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques|
|Publication status||Published - Jan 2006|