FRACTIONAL DIFFUSION LIMIT FOR A KINETIC FOKKER-PLANCK EQUATION WITH DIFFUSIVE BOUNDARY CONDITIONS IN THE HALF-LINE

• 成果类型: Article
• 署名作者: Bethencourt, Loic
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1683
• 发表日期: 2024
• 页码: 1713-1757
• 关键词: langevin process; anomalous diffusion; additive-functionals; THEOREMS

摘要

We consider a particle with position (X-t)(t >= 0) living in R+, whose velocity (V-t)(t >= 0) is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in (0, infinity). When it hits the boundary x = 0, the particle restarts with a random strictly positive velocity. We show that the properly rescaled position process converges weakly to a stable process reflected on its infimum. From a P.D.E. point of view, the time-marginals of (X-t, V-t)(t >= 0) solve a kinetic Fokker-Planck equation on (0, infinity) x R(+ )x R with diffusive boundary conditions. Properly rescaled, the space-marginal converges to the solution of some fractional heat equation on (0, infinity) x R+.