Current browse context:
math.PR
Change to browse by:
References & Citations
Mathematics > Probability
Title: Stochastic Nash evolution
(Submitted on 11 Dec 2023 (v1), last revised 25 Apr 2024 (this version, v2))
Abstract: This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given $\alpha \in ]\tfrac{1}{2},1]$, we show that a $C^{1,\alpha}$ 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; this is followed by a renormalization procedure for $C^{1,\alpha}$ embeddings. In particular, we also construct extrinsic Brownian motion when $g \in C^2$ and $u$ is a $C^{1,\alpha}$ isometric embedding.
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.
Submission history
From: Dominik Inauen [view email][v1] Mon, 11 Dec 2023 17:22:12 GMT (50kb)
[v2] Thu, 25 Apr 2024 12:01:23 GMT (52kb)
Link back to: arXiv, form interface, contact.