A non-local singular non-linear Fokker-Planck PDE
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| Format: | Preprint |
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2026
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| _version_ | 1866916004298227712 |
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| author | Bondi, Luca Issoglio, Elena Russo, Francesco |
| author_facet | Bondi, Luca Issoglio, Elena Russo, Francesco |
| contents | The focus of this paper is a non-local singular non-linear Fokker-Planck partial differential equation (PDE). The peculiarity of this PDE feature is in its divergence coefficient, which presents a product between a Besov distribution and a non-linearity. The latter involves the convolution between an integrable kernel K and the solution of the PDE, which leads to a non-locality of the first order term in the PDE. We prove existence and uniqueness of a solution to the PDE as well as continuity results on its coefficients. Previous analytical results are then applied to the study of well-posedness in law for a non-local singular McKean stochastic differential equation. As byproduct of that probabilistic representation, we establish mass conservation and positivity preserving for the PDE. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11801 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A non-local singular non-linear Fokker-Planck PDE Bondi, Luca Issoglio, Elena Russo, Francesco Probability The focus of this paper is a non-local singular non-linear Fokker-Planck partial differential equation (PDE). The peculiarity of this PDE feature is in its divergence coefficient, which presents a product between a Besov distribution and a non-linearity. The latter involves the convolution between an integrable kernel K and the solution of the PDE, which leads to a non-locality of the first order term in the PDE. We prove existence and uniqueness of a solution to the PDE as well as continuity results on its coefficients. Previous analytical results are then applied to the study of well-posedness in law for a non-local singular McKean stochastic differential equation. As byproduct of that probabilistic representation, we establish mass conservation and positivity preserving for the PDE. |
| title | A non-local singular non-linear Fokker-Planck PDE |
| topic | Probability |
| url | https://arxiv.org/abs/2605.11801 |