Intermittency and product of random matrices

Investigation of certain physical processes in a random medium can be reduced, under some assumption, to investigation of differential equations with random coefficients. As a rule, the relevant physical characteristics are expressed in terms of several first statistical moments of solutions and their calculation is of the main interest. Before this problem was formulated in a mathematical form, on a physical level there were predicted some spectacular features in behavior of solutions – one of them is a phenomenon of intermittency. Presence of intermittency complicates the estimation of the growth rates of moments substantially, because the main contribution to the growth of moments is carried by rare realizations with extreme values. It makes difficulties for interpretation of results of numerical modeling as well. On the other hand, in the mathematical theory of Furstenberg and Tutubalin, which were formulated in terms of product of random matrices, calculation of moments is equivalent to calculation of some invariant measure of generated Markovian chain. i.e. to solving of integral equation. Although this approach does not depend on facility of generators of random numbers, but this equation can be solved analytically in trivial cases only and for a long time there were no attempts to get a numerical solution even in a more or less simple relevant cases. In our report we will present, apparently, the first trial of numerical approach to the problem for the Jacobi equation and consider the first results for matrices 3x3, which are of the main practical interest.