Asymptotic behavior of a class of multiple time scales stochastic\n kinetic equations

2021/06/11 by Charles-Édouard Bréhier, Bréhier, Charles-Edouard, Rakotonirina-Ricquebourg, Shmuel +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2106.06417

openalex publication_date 2021/06/11 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider a class of stochastic kinetic equations, depending on two time\nscale separation parameters \ε and \δ: the evolution equation\ncontains singular terms with respect to \ε, and is driven by a fast\nergodic process which evolves at the time scale t/\δ². We prove that\nwhen (\ε,\δ)\→ (0,0) the density converges to the solution of a\nlinear diffusion PDE. This is a mixture of diffusion approximation in the PDE\nsense (with respect to the parameter \ε) and of averaging in the\nprobabilistic sense (with respect to the parameter \δ). The proof employs\nstopping times arguments and a suitable perturbed test functions approach which\nis adapted to consider the general regime \ε\≠ \δ.\n

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