Abstract:
The recursive and stochastic representation of solutions to the Fourier transformed
Navier-Stokes equations, as introduced by [34], is extended in several ways. First, associated
families of functions known as majorizing kernels are analyzed, in light of their
apparently essential role in the representation. Second, the theory is put on a more
comprehensive foundation by constructing the basic recursive object, the multiplicative
functional or its successor, the random field, without invoking the strong Markov property.
This allows the theory to embrace a wider class of evolutionary equations. Third, this
methodology, that has delivered global existence and uniqueness theorems for the Navier-
Stokes equations given suitably small initial datum, is extended to obtain local in time
existence and uniqueness results when the initial datum is arbitrarily large. Fourth, the
theory is applied to the semi-linear KPP equation linking it with, and extending previously
known results on the representation of solutions with branching Brownian motion.
