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

Motivated by the probabilistic methods for nonlinear differential equations introduced McKean (1975) Kolmogorov-Petrovski-Piskunov (KPP) equation, and Le Jan Sznitman (1997) 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 assumption that randomization intensities from generation is an ergodic time-reversible Markov process. In addition models arise in analysis certain deterministic this model includes multiplicative branching random walks, processes, generalizations percolation and/or cell aging Aldous Shields (1988) independently Athreya (1985).