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The paper explores the symbiotic relation between the Navier-Stokes equationsand the associated stochastic cascades. Specifically, we examine how somewell-known existence and uniqueness results for the Navier-Stokes equations caninform about the probabilistic features of the associated stochastic cascades,and how some probabilistic features of the stochastic cascades can, in turn,inform about the existence and uniqueness (or the lack thereof) of solutions.Our method of incorporating the stochastic explosion gives a simpler and morenatural method to construct the solution compared to the original constructionby Le Jan and Sznitman. This new stochastic construction is then used to showthe finite-time blowup and non-uniqueness of the initial value problem for theMontgomery-Smith equation. We exploit symmetry properties inherent in ourconstruction to give a simple proof of the global well-posedness results forsmall initial data in scale-critical Fourier-Besov spaces. We also obtain thepointwise convergence of the Picard's iteration associated with theFourier-transformed Navier-Stokes equations.

On Le Jan-Sznitman's stochastic approach to the Navier-Stokes equations