Title: Stochastic cascade solutions of the Navier-Stokes equations.
Abstract: Branching processes were used by McKean in 1975 to study the KPP-Fisher equation. His ideas were robust enough to apply to a larger class of semilinear parabolic equations. In 1997, Le Jan and Sznitman used similar ideas for the Navier-Stokes equations. They introduced a class of stochastic cascade solutions. Since then, the theory of cascade solutions has been quite fruitful, producing further insights on the uniqueness/nonuniqueness of solutions. Cascade solutions can also be defined for various toy models of the Navier-Stokes equations, for example the alpha-Riccati equation and the complex Burgers equation. In this talk, I will present some history and recent results of cascade solutions, with applications to the Navier-Stokes equations.