  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 {\it 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 chain. 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).
