Travelling waves solutions to the K-P-P equation at the critical wave speed: continuing Simon Harris' probabilistic analysis

Abstract

Recently Harris using probabilistic arguments alone has given new proofs of the known existence asymptotics and unique ness of travelling wave solutions to the KPP equation This paper is a sequel to Kyprianou b which provides alternative probabilistic arguments for supercritical wave speeds We complete our probabilis tic analysis here for the more dicult case of critical wave speeds The analysis is centered around the study of additive and multiplicative martingales and the construction of sizebiased measures on a space of nonhomogenous marked trees generated by a truncated branching Brownian motion As part of our results we also obtain a marti nale convergence theorem for the derivative of the additive martin gale Some of the main ideas are inspired by the techniques found in Kyprianou and Biggins and Lyons The value of these new probabilistic proofs is their generic nature which in principle can be generalized to study other types of spatial branching diusions and associated travelling waves