Stochastic Nash evolution
About ten years ago, De Lellis and Szekelyhidi made the surprising discovery that Nash’s results on the isometric embedding problem for Riemannian manifolds could be adapted to construct counterintuitive solutions to the Euler equations for incompressible flow. Their work shed new light on turbulence and nonlinear PDE. We use this link in the other direction, transferring ideas from turbulence to geometry.
A thermodynamic framework is introduced that connects two problems previously thought to be distinct: the isometric embedding problem for Riemannian manifolds and the construction of Brownian motion on Riemannian manifolds. This link is used to introduce a geometric stochastic flow that we term stochastic Nash evolution. The method has subtle ties and differences with Nash’s work.
This is joint work with Dominik Inauen (University of Leipzig).