{"paper":{"title":"One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Camille Tardif, Nicolas Fournier","submitted_at":"2018-05-24T15:26:12Z","abstract_excerpt":"We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like $(1+|v|)^{-\\beta}$ for some $\\beta>0$. We prove that, under a suitable rescaling, the position process resembles a Brownian motion if $\\beta\\geq 5$, a stable process if $\\beta\\in [1,5)$ and an integrated symmetric Bessel process if $\\beta\\in (0,1)$. The critical cases $\\beta=1$ and $\\beta=5$ require special rescaling. We recover some results of G.Lebeau and M.Puel [LP17], P.Cattiaux, E.Nasreddine and M. Puel [CNP16],"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09728","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}