K - P - P equation at the critical wave speed : continuing

Recently Harris (1999), using probabilistic arguments alone, has given new proofs of the (known) existence, asymptotics and uniqueness of travelling wave solutions to the K-P-P equation. This paper is a sequel to Kyprianou (2000b) which provides alternative probabilistic arguments for supercritical wave speeds. We complete our probabilis-tic analysis here for the more diicult case of critical wave speeds. The analysis is centered around the study of additive and multiplicative martingales and the construction of size-biased measures on a space of non-homogenous 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 (2000) and Lyons (1997). The value of these new probabilistic proofs is their generic nature which in principle can be generalized to study other types of spatial branching diiusions and associated travelling waves.