Viscosity Solutions of the Eikonal Equation on the Wasserstein Space
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913188056924160 |
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| author | Soner, H. Mete Yan, Qinxin |
| author_facet | Soner, H. Mete Yan, Qinxin |
| contents | Dynamic programming equations for mean field control problems with a separable structure are Eikonal equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to obtain a comparison result among sub and super solutions that are Lipschitz continuous in the Wasserstein-one norm which is directly verified for the value function. A key technical result provides a point-wise upper bound for the derivative of the linear derivative of a function in terms of its Lipschitz norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04097 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Viscosity Solutions of the Eikonal Equation on the Wasserstein Space Soner, H. Mete Yan, Qinxin Optimization and Control 35D40, 35Q89, 49L12, 49L25, 60G99 Dynamic programming equations for mean field control problems with a separable structure are Eikonal equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to obtain a comparison result among sub and super solutions that are Lipschitz continuous in the Wasserstein-one norm which is directly verified for the value function. A key technical result provides a point-wise upper bound for the derivative of the linear derivative of a function in terms of its Lipschitz norm. |
| title | Viscosity Solutions of the Eikonal Equation on the Wasserstein Space |
| topic | Optimization and Control 35D40, 35Q89, 49L12, 49L25, 60G99 |
| url | https://arxiv.org/abs/2308.04097 |