Stochastic Nash evolution

Dominik Inauen, Govind Menon

Published 2023-12-11Version 1

This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. We show that an embedding $u: M \to \mathbb{R}^q$ is isometric if and only if the intrinsic and extrinsic constructions of Brownian motion on $u(M)\subset \mathbb{R}^q$ yield processes with the same law. The equivalence is first established for smooth embeddings and it is followed by a renormalization procedure for $C^{1,\alpha}$ embeddings, $\alpha >\tfrac{1}{2}$. This formulation is based on a gedanken experiment that relates the intrinsic and extrinsic constructions of Brownian motion on an embedded manifold to the measurement of geodesic distance by observers in distinct frames of reference. This viewpoint provides a thermodynamic formalism for the isometric embedding problem that is suited to applications in geometric deep learning, stochastic optimization and turbulence.