Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918522219659264 |
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| author | Barbu, Viorel da Silva, José Luís Röckner, Michael |
| author_facet | Barbu, Viorel da Silva, José Luís Röckner, Michael |
| contents | This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $Ψ(-Δ)$, where $Ψ$ is a Bernstein function. As applications, the existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06388 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems Barbu, Viorel da Silva, José Luís Röckner, Michael Analysis of PDEs Probability 60H15, 47H05, 47J05 This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $Ψ(-Δ)$, where $Ψ$ is a Bernstein function. As applications, the existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs. |
| title | Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems |
| topic | Analysis of PDEs Probability 60H15, 47H05, 47J05 |
| url | https://arxiv.org/abs/2308.06388 |