Fractional kinetic equations

TL;DR

This paper introduces a new class of fractional kinetic equations derived from non-Markovian continuous time random walk approximations, providing well-posedness and probabilistic solutions for interacting particle systems with variable order.

Contribution

It develops a general framework for fractional kinetic measure-valued evolutions with variable order, extending previous Markovian models to non-Markovian, fractional dynamics.

Findings

Proves well-posedness of fractional kinetic equations.

Provides probabilistic formulas for solutions.

Focuses on fractional interacting diffusions.

Abstract

We develop the idea of non-Markovian CTRW (continuous time random walk) approximation to the evolution of interacting particle systems, which leads to a general class of fractional kinetic measure-valued evolutions with variable order. We prove the well-posedness of the resulting new equations and present a probabilistic formula for their solutions. Though our method are quite general, for simplicity we treat in detail only the fractional versions of the interacting diffusions. The paper can be considered as a development of the ideas from the works of Belavkin and Maslov devoted to Markovian (quantum and classical) systems of interacting particles.