Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.06541 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917649448960000 |
|---|---|
| author | Inauen, Dominik Menon, Govind |
| author_facet | Inauen, Dominik Menon, Govind |
| contents | This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given $α\in ]\tfrac{1}{2},1]$, we show that a $C^{1,α}$ 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 $C^{1,α}$ embeddings. In particular, we also construct extrinsic Brownian motion when $g \in C^2$ and $u$ is a $C^{1,α}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06541 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stochastic Nash evolution Inauen, Dominik Menon, Govind Probability Mathematical Physics Analysis of PDEs 57N35, 60J65 This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given $α\in ]\tfrac{1}{2},1]$, we show that a $C^{1,α}$ 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 $C^{1,α}$ embeddings. In particular, we also construct extrinsic Brownian motion when $g \in C^2$ and $u$ is a $C^{1,α}$ 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. |
| title | Stochastic Nash evolution |
| topic | Probability Mathematical Physics Analysis of PDEs 57N35, 60J65 |
| url | https://arxiv.org/abs/2312.06541 |