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

TL;DR

This paper investigates the explosion behavior of doubly stochastic Yule cascades, a class of stochastic models relevant to PDE solutions and percolation phenomena, providing criteria for explosion and non-explosion without assuming reversibility.

Contribution

It extends previous work by establishing explosion criteria for non-reversible doubly stochastic Yule cascades using new analytical techniques.

Findings

Criteria for explosion and non-explosion established

Application to Navier-Stokes cascade in 3D confirmed explosion

Non-explosion results for high-dimensional cases (d ≥ 12)

Abstract

We analyze the explosion problem for a class of stochastic models introduced in Part I (arXiv:2103.06912), referred to as doubly stochastic Yule cascades. These models arise naturally in the construction of solutions to evolutionary PDEs as well as in purely probabilistic first passage percolation phenomena having a Markov-type statistical dependence, new for this context. Using cut-set arguments and a greedy algorithm, we respectively establish criteria for non-explosion and explosion without requiring the time-reversibility of the underlying branching Markov chain (a condition required in Part I). Notable applications include the explosion of the self-similar cascade of the Navier-Stokes equations in dimension $d=3$ and non-explosion in dimensions $d\ge 12$.