%0 Journal Article %1 MR1617047 %A Gall, Jean-Francois Le %A Jan, Yves Le %D 1998 %J Ann. Probab. %K LeGall Levy_processes Levy_trees branching_processes continuous-state_branching_processes exploration_process %N 1 %P 213--252 %T Branching processes in Lévy processes: the exploration process %V 26 %X It has been known for a long time that there exist fundamental connections between branching trees and random walks (or processes with stationary independent increments). Early papers are due to Harris (1952), D. G. Kendall (1951), Dwass (1969) and Lamperti (1967) (they are cited in the paper). The present paper by Le Gall and Le Jan is a further important step to a deeper understanding of this connection. The basic notions are the exploration process of a Galton-Watson tree and the height process of a Lévy process. The exploration process at time $n$ is the generation of the individual in the tree visited at time $n$, assuming that the individuals are visited successively to the law of primogeniture. This process is in itself complicated, but it can be constructed as a functional of a suitable left-continuous random walk. Conversely, the authors construct the height process of a spectrally positive Lévy process, which can be understood as the exploration process of a general continuous branching process. This construction has important applications. In particular, it allows an adequate formulation of the classical Ray-Knight theorem for such Lévy processes. For related results see a paper by J. Geiger in Stochastic partial differential equations (Edinburgh, 1994), 72--96, Cambridge Univ. Press, Cambridge, 1995; MR1352736 (96f:60146).

@article{MR1617047, abstract = {It has been known for a long time that there exist fundamental connections between branching trees and random walks (or processes with stationary independent increments). Early papers are due to Harris (1952), D. G. Kendall (1951), Dwass (1969) and Lamperti (1967) (they are cited in the paper). The present paper by Le Gall and Le Jan is a further important step to a deeper understanding of this connection. The basic notions are the exploration process of a Galton-Watson tree and the height process of a Lévy process. The exploration process at time $n$ is the generation of the individual in the tree visited at time $n$, assuming that the individuals are visited successively to the law of primogeniture. This process is in itself complicated, but it can be constructed as a functional of a suitable left-continuous random walk. Conversely, the authors construct the height process of a spectrally positive L\'{e}vy process, which can be understood as the exploration process of a general continuous branching process. This construction has important applications. In particular, it allows an adequate formulation of the classical Ray-Knight theorem for such L\'{e}vy processes. For related results see a paper by J. Geiger [in Stochastic partial differential equations (Edinburgh, 1994), 72--96, Cambridge Univ. Press, Cambridge, 1995; MR1352736 (96f:60146)]. }, added-at = {2009-01-15T22:21:43.000+0100}, author = {Gall, Jean-Francois Le and Jan, Yves Le}, biburl = {https://www.bibsonomy.org/bibtex/2db9eafa589a0062780fb32b84a207e0d/peter.ralph}, coden = {APBYAE}, description = {Branching processes in {L}\'evy processes: the exploration process -- Le Gall and Le Jan}, fjournal = {The Annals of Probability}, interhash = {5ea21cc55b25aaea80dfbb41eacee753}, intrahash = {db9eafa589a0062780fb32b84a207e0d}, issn = {0091-1798}, journal = {Ann. Probab.}, keywords = {LeGall Levy_processes Levy_trees branching_processes continuous-state_branching_processes exploration_process}, mrclass = {60J80 (60J25 60J30 60J55)}, mrnumber = {MR1617047 (99d:60096)}, mrreviewer = {G{\"o}tz Kersting}, number = 1, pages = {213--252}, timestamp = {2009-01-15T22:21:43.000+0100}, title = {Branching processes in {L}\'evy processes: the exploration process}, volume = 26, year = 1998 }