An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909319068385280 |
|---|---|
| author | Bertucci, Charles Lions, Pierre Louis |
| author_facet | Bertucci, Charles Lions, Pierre Louis |
| contents | We provide a simple $C^{1,1}$ approximation of the squared Wasserstein distance on R^d when one of the two measures is fixed. This approximation converges locally uniformly. More importantly, at points where the differential of the squared Wasserstein distance exists, it attracts the differentials of the approximations at nearby points. Our method relies on the Hilbertian lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and Lions. We then provide an application of this result by using it to establish a comparison principle for an Hamilton-Jacobi equation on the set of probability measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations Bertucci, Charles Lions, Pierre Louis Analysis of PDEs Optimization and Control Probability We provide a simple $C^{1,1}$ approximation of the squared Wasserstein distance on R^d when one of the two measures is fixed. This approximation converges locally uniformly. More importantly, at points where the differential of the squared Wasserstein distance exists, it attracts the differentials of the approximations at nearby points. Our method relies on the Hilbertian lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and Lions. We then provide an application of this result by using it to establish a comparison principle for an Hamilton-Jacobi equation on the set of probability measures. |
| title | An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations |
| topic | Analysis of PDEs Optimization and Control Probability |
| url | https://arxiv.org/abs/2409.11793 |