Self-similarity and diffusive limits for linear kinetic equations: a Wild sum approach
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| Format: | Preprint |
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2026
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| _version_ | 1866910252275859456 |
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| author | Cañizo, José A. Mischler, Stéphane Tassi, Niccolò |
| author_facet | Cañizo, José A. Mischler, Stéphane Tassi, Niccolò |
| contents | We prove that linear collisional kinetic equations in the whole space without confinement mechanism display a long-time self-similar behaviour.
This drastically improves the recently known results (decay estimates) about the solutions in such a context, providing the first result regarding this self-similar behaviour. As a consequence, we also establish a uniform-in-time convergence of the suitably rescaled solutions to their diffusion limit, which is also new.
The class of equations considered includes some BGK type equations, some kinetic nonlocal Fokker--Planck-type equations and
some kinetic (possibly fractional) Fokker--Planck equations, for which we are able to write explicitly solutions through a Wild sum (or Dyson series)
or we can manage some accurate computations on the Fourier side. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24685 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-similarity and diffusive limits for linear kinetic equations: a Wild sum approach Cañizo, José A. Mischler, Stéphane Tassi, Niccolò Analysis of PDEs Probability 35B40 (Primary) 82C40, 35R11, 45M05, 60E15, 60K35 (Secondary) We prove that linear collisional kinetic equations in the whole space without confinement mechanism display a long-time self-similar behaviour. This drastically improves the recently known results (decay estimates) about the solutions in such a context, providing the first result regarding this self-similar behaviour. As a consequence, we also establish a uniform-in-time convergence of the suitably rescaled solutions to their diffusion limit, which is also new. The class of equations considered includes some BGK type equations, some kinetic nonlocal Fokker--Planck-type equations and some kinetic (possibly fractional) Fokker--Planck equations, for which we are able to write explicitly solutions through a Wild sum (or Dyson series) or we can manage some accurate computations on the Fourier side. |
| title | Self-similarity and diffusive limits for linear kinetic equations: a Wild sum approach |
| topic | Analysis of PDEs Probability 35B40 (Primary) 82C40, 35R11, 45M05, 60E15, 60K35 (Secondary) |
| url | https://arxiv.org/abs/2605.24685 |