A note on branching Lévy processes

We show for the branching Lévy process that it is possible to construct two classes of multiplicative martingales using stopping lines and solutions to one of two source equations. The first class, similar to those martingales of Chauvin (1991, Ann. Probab. 30, 1195–1205) and Neveu (1988, Seminar on Stochastic Processes 1987, Progress in Probability and Statistics, vol. 15, Birkhaüser, Boston, pp. 223–241) have a source equation which provides travelling wave solutions to a generalized version of the K-P-P equation. For the second class of martingales, similar to those of Biggins and Kyprianou (1997, Ann. Probab. 25, 337–360), the source equation is a functional equation. We show further that under reasonably broad circumstances, these equations share the same solutions and hence the two types of martingales are one and the same. This conclusion also tells us something more about the nature of the solutions to the first of our two equations.