Stochastic Nash evolution

We introduce a probabilistic formulation for the Nash embedding theorems. Our approach inverts the usual relation between mathematics and physics. We use rigorous mathematical results, including Nash’s work, results of De Lellis and Szekelyhidi, and work of the speaker and Rezakhanlou, to guide the design of algorithms and evolution equations.

We use relaxation as in Nash’s work, but replace his iteration (in low codimension) or continuous flow (in high codimension) with a stochastic flow. The main issue in the derivation of our flow is a principled resolution of a semidefinite program. The same fundamental structure applies to several hard constraint systems and nonlinear PDE.