Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis

Reference:

Kyprianou, A. E., 2004. Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis. Annales De L Institut Henri Poincare-Probabilites Et Statistiques, 40 (1), pp. 53-72.

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Official URL:

http://dx.doi.org/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

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