Self-similar solutions of kinetic-type equations: The boundary case

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Bogus, Kamil¹  Buraczewski, Dariusz²  Marynych, Alexander³ 

[1] Wroclaw Univ Sci & Technol, Fac Pure & Appl Math, Ul Wybrzeze Wyspianskiego 27, PL-50370 Wroclaw, Poland

[2] Univ Wroclaw, Math Inst, Pl Grunwaldzki 2-4, PL-50384 Wroclaw, Poland

[3] Taras Shevchenko Natl Univ Kyiv, Fac Comp Sci & Cybernet, UA-01601 Kiev, Ukraine

关键词: Biggins martingale;    Derivative martingale;    Kac model;    Kinetic equation;    Random trees;    Smoothing transform;   

DOI  :  10.1016/j.spa.2019.03.005

来源: Elsevier

【 摘 要 】

For a time dependent family of Probability measures (rho(t))(t >= 0) we consider a kinetic-type evolution equation partial derivative phi(t)/partial derivative t + phi(t) = (Q) over cap phi(t) where (Q) over cap is a smoothing transform and phi(t) is the Fourier-Stieltjes transform of rho(t). Assuming that the initial measure rho(0) belongs to the domain of attraction of a stable law, we describe asymptotic properties of rho(t), as t -> infinity. We consider the boundary regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures (rho(t))(t >= 0) that refines the corresponding construction proposed in Bassetti and Ladelli, (2012). (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free