Nonlinear rough Fokker-Planck equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915406306869248 |
|---|---|
| author | Bugini, Fabio Friz, Peter K. Stannat, Wilhelm |
| author_facet | Bugini, Fabio Friz, Peter K. Stannat, Wilhelm |
| contents | McKean-Vlasov SDEs describe systems where the dynamics depend on the law of the process. The corresponding Fokker-Planck equation is a nonlinear, nonlocal PDE for the corresponding measure flow. In the presence of common noise and conditional law dependence, the evolution becomes random and is governed by a stochastic Fokker-Planck equation; that is, a nonlinear, nonlocal SPDE in the space of measures. (Such equations constitute an important ingredient in the theory of mean-field games with common noise.) Well-posedness of such SPDEs is a difficult problem, the best result to date due to Coghi-Gess (2019), which however comes with dimension-dependent regularity assumptions. In the present work, we show how rough path techniques can circumvent these entirely. Hence, and somewhat contrarily to common believe, the use of rough paths leads to substantially less regularity demands on the coefficients than methods rooted in classical stochastic analysis methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17469 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear rough Fokker-Planck equations Bugini, Fabio Friz, Peter K. Stannat, Wilhelm Probability 60L50, 35Q84 McKean-Vlasov SDEs describe systems where the dynamics depend on the law of the process. The corresponding Fokker-Planck equation is a nonlinear, nonlocal PDE for the corresponding measure flow. In the presence of common noise and conditional law dependence, the evolution becomes random and is governed by a stochastic Fokker-Planck equation; that is, a nonlinear, nonlocal SPDE in the space of measures. (Such equations constitute an important ingredient in the theory of mean-field games with common noise.) Well-posedness of such SPDEs is a difficult problem, the best result to date due to Coghi-Gess (2019), which however comes with dimension-dependent regularity assumptions. In the present work, we show how rough path techniques can circumvent these entirely. Hence, and somewhat contrarily to common believe, the use of rough paths leads to substantially less regularity demands on the coefficients than methods rooted in classical stochastic analysis methods. |
| title | Nonlinear rough Fokker-Planck equations |
| topic | Probability 60L50, 35Q84 |
| url | https://arxiv.org/abs/2507.17469 |