THE MULTIPLICATIVE INTEGRAL AND THE EVOLUTION OF THE MAGNETIC FIELD IN THE MARKOV LINEAR MODEL

Abstract

The paper is devoted to the probabilistic asymptotic analysis of the magnetic field and magnetic energy in a Markov linear model of an incompressible fluid. Firstly, the paper introduces a brief history of the problem under consideration and presents the main results of the previous studies, which ultimately lead to the study of the product of independent random matrices with an increasing number of multiplicands. After that, the description of the Markov linear model considered in the paper is given, the so-called Lyapunov (generally speaking, random) bases for the multiplicative (stochastic) integral contained in the integral representation of the magnetic field are constructed. In conclusion, by decomposing the multiplicative integral over the constructed Lyapunov basis and relying on the properties of the basis, the main results - theorems on the asymptotic behavior of the magnetic field and magnetic energy - have been proven.