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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2203.11701 |
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| _version_ | 1866910602172039168 |
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| author | Gigli, Nicola Tamanini, Luca Trevisan, Dario |
| author_facet | Gigli, Nicola Tamanini, Luca Trevisan, Dario |
| contents | The aim of this paper is twofold.
- In the setting of RCD(K,$\infty$) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the Hamilton--Jacobi equation. We use these estimates to prove that, as the viscosity tends to zero, solutions of this equation converge to the evolution driven by the Hopf--Lax formula, in accordance with the smooth case.
- We then use such convergence to study the small-time Large Deviation Principle for both the heat kernel and the Brownian motion: we obtain the expected behavior under the additional assumption that the space is proper.
As an application of the latter point, we also discuss the $Γ$-convergence of the Schrödinger problem to the quadratic optimal transport problem in proper RCD(K,$\infty$) spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11701 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations Gigli, Nicola Tamanini, Luca Trevisan, Dario Probability Metric Geometry The aim of this paper is twofold. - In the setting of RCD(K,$\infty$) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the Hamilton--Jacobi equation. We use these estimates to prove that, as the viscosity tends to zero, solutions of this equation converge to the evolution driven by the Hopf--Lax formula, in accordance with the smooth case. - We then use such convergence to study the small-time Large Deviation Principle for both the heat kernel and the Brownian motion: we obtain the expected behavior under the additional assumption that the space is proper. As an application of the latter point, we also discuss the $Γ$-convergence of the Schrödinger problem to the quadratic optimal transport problem in proper RCD(K,$\infty$) spaces. |
| title | Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2203.11701 |