Article,
Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis
Annales de l'Institut Henri Poincare (B) Probability and Statistics, 40 (1): 53--72 (2004)
DOI: DOI: 10.1016/j.anihpb.2003.06.001
Abstract
Recently Harris Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503, using probabilistic methods alone, has given new proofs for the existence, asymptotics and uniqueness of travelling wave solutions to the K-P-P equation. Following in this vein we outline alternative probabilistic proofs. Specifically the techniques are confined to the study of additive and multiplicative martingales and spinal path decompositions along the lines of B. Chauvin, A. Rouault, Probab. Theory Related Fields 80 (1988) 299, R. Lyons, in: K.B. Athreya, P. Jagers (eds.), Classical and Modern Branching Processes, Vol. 84, Springer-Verlag, New York, 1997, pp. 217-222 and R. Lyons et al., Ann. Probab. 23 (1995) 1125. We also make use of a new decomposition where the spine is a conditioned process. Some new results concerning martingale convergence are obtained as a by-product of the analysis.
