$χ^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914528005980160 |
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| author | Avelin, Benny Barrera, Gerardo |
| author_facet | Avelin, Benny Barrera, Gerardo |
| contents | In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin--Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated $χ^2$-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker--Planck equations associated to the Langevin--Kolmogorov dynamics. In addition, this explicit analysis yields asymptotics of the mixing times and, in some regimes, information on the limiting profile, going beyond the product condition and the cut-off window alone. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29473 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $χ^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials Avelin, Benny Barrera, Gerardo Probability Classical Analysis and ODEs Dynamical Systems 37A25, 60H10, 34D10, 82C31, 34E20 In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin--Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated $χ^2$-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker--Planck equations associated to the Langevin--Kolmogorov dynamics. In addition, this explicit analysis yields asymptotics of the mixing times and, in some regimes, information on the limiting profile, going beyond the product condition and the cut-off window alone. |
| title | $χ^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials |
| topic | Probability Classical Analysis and ODEs Dynamical Systems 37A25, 60H10, 34D10, 82C31, 34E20 |
| url | https://arxiv.org/abs/2603.29473 |