Some probability theory that arises from worrying about Navier-Stokes and other quasilinear equations

The success of probability theory in the analysis of linear, or even certain semi-linear, parabolic and elliptic pde’s is well documented. In spite of various attempts to find a stochastic foothold for the analysis of Navier-Stokes equations and related quasilinear equations, the problem remains a substantial challenge. That said, the quest can lead to new stochastic structures and problems that relate to modern probability in fundamental ways. In this talk I will try to indicate this with a few explicit examples largely stemming from the Lejan-Sznitman multiplicative cascade/branching random walk framework for Navier-Stokes equations.

This talk is primarily based on recent joint work with Radu Dascaliuc, Nicholas Michalowski, and Enrique Thomann with partial support from the National Science Foundation.