A non-explosion criterion for branching Markov chains and applications
In 1975, McKean used a stochastic cascade to construct a solution to the KPP equation in the physical domain. In 1997, Le Jan and Sznitmann used a similar approach for the Navier-Stokes Equations (NSE) in the Fourier domain. Their construction leads to a problem known as stochastic explosion: will the cascade reach every finite horizon within finitely many steps (non-explosion), or can it happen that there will be infinitely many branches in a finite horizon (explosion)? This problem has been studied for the cascade associated with the NSE, the complex Burgers equation, and the alpha-Ricatti equation. In these equations, non-explosion is equivalent to the uniqueness of solutions, which may be resolved by an analytic approach. From a probabilistic perspective, however, the problem of stochastic explosion can be formulated generally in terms of a branching Markov chain and an independent family of holding times without connection with a PDE. We use “cutset” arguments to give a criterion for non-explosion. Applications include the KPP equation in Fourier domain, the Bessel cascade of NSE, and the Yule process. In this talk, I will give a synopsis of this approach. Joint work with Radu Dascaliuc, Enrique Thomann and Edward Waymire.