A New Approach to Fractional Kinetic Evolutionsстатья
Статья опубликована в высокорейтинговом журнале
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Дата последнего поиска статьи во внешних источниках: 8 июня 2022 г.
Аннотация:Kinetic equations describe the limiting deterministic evolution of properly scaled systemsof interacting particles. A rather universal extension of the classical evolutions, that aims to take intoaccount the effects of memory, suggests the generalization of these evolutions obtained by changingthe standard time derivative with a fractional one. In the present paper, extending some previousnotes of the authors related to models with a finite state space, we develop systematically the idea ofCTRW (continuous time random walk) modelling of the Markovian evolution of interacting particlesystems, which leads to a more nontrivial class of fractional kinetic measure-valued evolutions, withthe mixed fractional order derivatives varying with the change of the state of the particle system, andwith variational derivatives with respect to the measure variable. We rigorously justify the limitingprocedure, prove the well-posedness of the new equations, and present a probabilistic formula fortheir solutions. As the most basic examples we present the fractional versions of the Smoluchovskicoagulation and Boltzmann collision models.