Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917236899315712 |
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| author | Tassi, Niccolò |
| author_facet | Tassi, Niccolò |
| contents | We investigate the asymptotic behaviour of solutions of a class of nonlocal Fokker--Planck equations defined by nonsingular, heavy-tailed convolution kernels and characterised by a scaling parameter $\e\in(0,1]$ and a fractional index $s\in(1/2,1)$. By employing a suitable version of the generalised central limit for heavy-tailed distributions and the use of Harris's theorem, we prove exponential convergence to the equilibrium with a rate that is independent of both $\e$ and $s$. This allows us to show uniform--in--time convergence for both $\e\to 0$ and $s\to1$ recovering the limiting equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00375 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel Tassi, Niccolò Analysis of PDEs Probability 35B40, 45K05 (Primary) 35C10, 26A33, 35C10, 60F05 (Secondary) We investigate the asymptotic behaviour of solutions of a class of nonlocal Fokker--Planck equations defined by nonsingular, heavy-tailed convolution kernels and characterised by a scaling parameter $\e\in(0,1]$ and a fractional index $s\in(1/2,1)$. By employing a suitable version of the generalised central limit for heavy-tailed distributions and the use of Harris's theorem, we prove exponential convergence to the equilibrium with a rate that is independent of both $\e$ and $s$. This allows us to show uniform--in--time convergence for both $\e\to 0$ and $s\to1$ recovering the limiting equations. |
| title | Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel |
| topic | Analysis of PDEs Probability 35B40, 45K05 (Primary) 35C10, 26A33, 35C10, 60F05 (Secondary) |
| url | https://arxiv.org/abs/2602.00375 |