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arXiv · 2107.13182

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

Radu Dascaliuc·Tuan N. Pham·Enrique Thomann·Edward C. Waymire
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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 $d\ge 12$.

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Radu Dascaliuc, Tuan N. Pham, Enrique Thomann, Edward C. Waymire. 2021-07-28. Doubly stochastic Yule cascades (Part II): The explosion problem in the non-reversible case. https://arxiv.org/abs/2107.13182

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