Further probabilistic 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 −ρ, perform dyadic branching at rate β 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 −ρ with solutions f : [0,∞) → [0, 1] satisfying f(0) = 1 and f(∞) = 0 has a unique solution with a particular asymptotic when ρ < √ 2β, and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of Harris [18] and Kyprianou [27] 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 √ 2β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, Wλ, 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 λ ∈ (0, √ 2β − ρ). Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > √ 2β − ρ. This result combined with Chauvin and Rouault’s [9] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals Wλ as the limiting Radon-Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. ∗Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, U.K.. email: [email protected] †Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, U.K.. email: [email protected] ‡Mathematical institute, Utrecht University, P.O.Box 80.010, 3508 TA, Utrecht, The Netherlands. email: [email protected]