Wasserstein Diffusion on Multidimensional Spaces
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916220220997632 |
|---|---|
| author | Sturm, Karl-Theodor |
| author_facet | Sturm, Karl-Theodor |
| contents | Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is
(i) reversible w.r.t.~the entropic measure ${\mathbb P}^β$ on ${\mathcal P}(M)$, heuristically given as $$d\mathbb{P}^β(μ)=\frac{1}{Z} e^{-β\, \text{Ent}(μ| m)}\ d\mathbb{P}^*(μ);$$ (ii) associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\mathcal E}_W(f)=\liminf_{g\to f}\ \frac12\int_{{\mathcal P}(M)} \big\|\nabla_W g\big\|^2(μ)\ d{\mathbb P}^β(μ);$$ (iii) non-degenerate, at least in the case of the $n$-sphere and the $n$-torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12721 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wasserstein Diffusion on Multidimensional Spaces Sturm, Karl-Theodor Probability Functional Analysis Metric Geometry Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^β$ on ${\mathcal P}(M)$, heuristically given as $$d\mathbb{P}^β(μ)=\frac{1}{Z} e^{-β\, \text{Ent}(μ| m)}\ d\mathbb{P}^*(μ);$$ (ii) associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\mathcal E}_W(f)=\liminf_{g\to f}\ \frac12\int_{{\mathcal P}(M)} \big\|\nabla_W g\big\|^2(μ)\ d{\mathbb P}^β(μ);$$ (iii) non-degenerate, at least in the case of the $n$-sphere and the $n$-torus. |
| title | Wasserstein Diffusion on Multidimensional Spaces |
| topic | Probability Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2401.12721 |