  The paper explores the symbiotic relation between the Navier-Stokes equations
and the associated stochastic cascades. Specifically, we examine how some
well-known existence and uniqueness results for the Navier-Stokes equations can
inform 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 more
natural method to construct the solution compared to the original construction
by Le Jan and Sznitman. This new stochastic construction is then used to show
the finite-time blowup and non-uniqueness of the initial value problem for the
Montgomery-Smith equation. We exploit symmetry properties inherent in our
construction to give a simple proof of the global well-posedness results for
small initial data in scale-critical Fourier-Besov spaces. We also obtain the
pointwise convergence of the Picard's iteration associated with the
Fourier-transformed Navier-Stokes equations.
