Some applications of matrix continued fractions
for probable description of brownian motions
O.V. Muzychuk
N. Novgorod Arch. and Siv.Eng.Acad., Russia
As we know, dynamic systems subjected to the action of random forces, are usualy discribed using the apparatus of Markov processes. However, analytic solutions of corresponding Fokker-Planck equations have been found in individual cases only. Mathematical difficulties increase considerably if the random actions cannot be approximated by delta-correlated noise. The same situation takes place in the Brownian motion problems.
In the number of cases one can use matrix continued fractions to find the statistical moments of the solution. Basic principle of this method consists of introducing vectors, whose components are combined moments or combined cumulants of the stochastic solution and the random force. In some important situations the linking of these vectors, due to the non-linearity, takes the form of a three-term interaction and the required moments take the form of matrix continued fractions. Corresponding calculational procedure one can realise by computer. For one-dimensional Brownian motion under the action of Gaussian delta- correlated noise matrix fractions reduse to ordinary continued fractions.
For example some results for relaxation of the variance and 4-th cumulant of one-dimensional Brownian motion have been obtained. One can use this method to inverstigate probable characteristices of the motion subjected to the action of random potential.