Variational structures for the Fokker--Planck equation with general Dirichlet boundary conditions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914193098145792 |
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| author | Quattrocchi, Filippo |
| author_facet | Quattrocchi, Filippo |
| contents | We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $Ω\Subset \mathbb R^d$ with general -- strictly positive and temporally constant -- Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum.
In the special case where $Ω$ is an interval in $\mathbb R^1$, we prove that such a solution is a gradient flow -- curve of maximal slope -- within a suitable space of measures, endowed with a modified Wasserstein distance.
Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\partial Ω$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure $\overline Ω$.
The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $Ω$ is an interval in $\mathbb R^1$, we find a formula for the descending slope of this geodesically nonconvex functional. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variational structures for the Fokker--Planck equation with general Dirichlet boundary conditions Quattrocchi, Filippo Analysis of PDEs Metric Geometry Optimization and Control 49Q20, 49Q22, 35A15, 35K20, 35Q84 We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $Ω\Subset \mathbb R^d$ with general -- strictly positive and temporally constant -- Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where $Ω$ is an interval in $\mathbb R^1$, we prove that such a solution is a gradient flow -- curve of maximal slope -- within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\partial Ω$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure $\overline Ω$. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $Ω$ is an interval in $\mathbb R^1$, we find a formula for the descending slope of this geodesically nonconvex functional. |
| title | Variational structures for the Fokker--Planck equation with general Dirichlet boundary conditions |
| topic | Analysis of PDEs Metric Geometry Optimization and Control 49Q20, 49Q22, 35A15, 35K20, 35Q84 |
| url | https://arxiv.org/abs/2403.07803 |