Solutions of some partial differential equations have representations as expectations of stochastic processes. These representations can give insight into the existence and uniqueness of solutions. This talk will open with a short overview of the construction of stochastic representations of solutions and then discuss applications of this methodology to the Navier-Stokes equations governing the velocity of incompressible fluids in 3 dimensions.
Although the Navier-Stokes equations have been studied extensively, important questions remain concerning the existence and uniqueness of solutions. Descriptions will be given of two types of representations, both formulated in terms of conditional expectations of functionals of branching semi-Markov processes. The first type, originally developed by LeJan and Sznitman in (1997), yields representations of Fourier-transformed solutions. The other type yields physical space representations of solutions. Both give existence and uniqueness of solutions for all time for `small' initial data and on short time intervals for `large' initial data.