Abstract:
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.
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