We propose a trajectorial construction of a class of measure-valued Markov processes, called superprocesses or measure-valued branching processes, which have been studied extensively in the last few years. These processes were originally defined as weak limits of systems of branching particles. The basic idea of our construction is to use the branching structure of excursions of a linear Brownian motion to model the branching mechanism of the superprocess. Without any additional effort, our approach leads to the so-called historical process, which contains more information than the superprocess in the sense that it keeps track of the individual paths followed by the particles. We emphasize the relationship between the properties of the historical process and the corresponding results of excursion theory. We also give a description of the support of the superprocess at a fixed time, using a simple tree model. Finally, we use our construction to recover certain pathwise properties recently obtained by Perkins.
Jean-Francois Le Gall. "Brownian Excursions, Trees and Measure-Valued Branching Processes." Ann. Probab. 19 (4) 1399 - 1439, October, 1991. https://doi.org/10.1214/aop/1176990218