From exploration paths to mass excursions – variations on a theme of Ray and Knight

Abstract

Based on an intuitive approach to the Ray–Knight representation of Feller’s branching diffusion in terms of Brownian excursions we survey a few recent developments around exploration and mass excursions. One of these is Bertoin’s “tree of alleles with rare mutations” [6], seen as a tree of excursions of Feller’s branching diffusion. Another one is a model of a population with individual reproduction, pairwise fights and emigration to ever new colonies, conceived as a tree of excursions of Feller’s branching diffusion with logistic growth [14]. Finally, we report on a Ray–Knight representation of Feller’s branching diffusion with logistic growth in terms of a reflected Brownian motion whose drift depends on the local time accumulated at its current level [19].