Strong solutions to McKean-Vlasov SDEs associated to a class of degenerate Fokker-Planck equations with coefficients of Nemytskii-type
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| Format: | Preprint |
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2024
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| _version_ | 1866910437954551808 |
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| author | Grube, Sebastian |
| author_facet | Grube, Sebastian |
| contents | While the nondegenerate case is well-known, there are only few results on the existence of strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker-Planck(-Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from $L^1$, and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point $x\in\mathbb{R}$. For these kind of equations the existence of a Schwartz-distributional solution $u$ is well-known. We show that there exists a unique strong solution to the associated degenerate McKean-Vlasov SDE with time marginal law densities $u$. In particular, every weak solution to this equation with time marginal law densities $u$ can be written as a functional of the driving Brownian motion.
Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities $u$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_18918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong solutions to McKean-Vlasov SDEs associated to a class of degenerate Fokker-Planck equations with coefficients of Nemytskii-type Grube, Sebastian Probability Analysis of PDEs 60H10, 60G17, 35C99 While the nondegenerate case is well-known, there are only few results on the existence of strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker-Planck(-Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from $L^1$, and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point $x\in\mathbb{R}$. For these kind of equations the existence of a Schwartz-distributional solution $u$ is well-known. We show that there exists a unique strong solution to the associated degenerate McKean-Vlasov SDE with time marginal law densities $u$. In particular, every weak solution to this equation with time marginal law densities $u$ can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities $u$. |
| title | Strong solutions to McKean-Vlasov SDEs associated to a class of degenerate Fokker-Planck equations with coefficients of Nemytskii-type |
| topic | Probability Analysis of PDEs 60H10, 60G17, 35C99 |
| url | https://arxiv.org/abs/2404.18918 |