Doubly Stochastic Yule Cascades (Part I): The explosion problem in the time-reversible case

TL;DR

This paper introduces Doubly Stochastic Yule cascades, a new class of stochastic models, and provides criteria to prevent explosion in the case of ergodic, time-reversible intensity randomization, connecting to various branching processes.

Contribution

It defines a novel class of stochastic cascade models with non-explosion criteria under ergodic, time-reversible intensity randomization, extending existing branching process frameworks.

Findings

Established non-explosion criteria for the models.

Connected the cascade models to known branching processes.

Included models relevant to nonlinear differential equations.

Abstract

Motivated by the probabilistic methods for nonlinear differential equations introduced by McKean (1975) for the Kolmogorov-Petrovski-Piskunov (KPP) equation, and by Le Jan and Sznitman (1997) for the incompressible Navier-Stokes equations, we identify a new class of stochastic cascade models, referred to as Doubly Stochastic Yule cascades. We establish non-explosion criteria under the assumption that the randomization of Yule intensities from generation to generation is by an ergodic time-reversible Markov process. In addition to the cascade models that arise in the analysis of certain deterministic nonlinear differential equations, this model includes the multiplicative branching random walks, the branching Markov processes, and the stochastic generalizations of the percolation and/or cell aging models introduced by Aldous and Shields (1988) and independently by Athreya (1985).