A multiwave transmission line with random non-homogeneities and a Brownian movement in Siegel's circle

A multiwave transmission line without loses is considered. After a similarity transformation of the matrix coefficient of reflection, it becomes a point of the classical matrix, domain of the first kind, in other words, Siegel's circle.
A transmission along the transmission line leads to a linear fractional transformation of Siegel's circle onto itself. A diffusion equation for a random walk corresponding to these transformations in Siegel's circle is obtained. The invariance of the diffusuion equation enables to study the statistics of the random distance from zero matrix to a walkingspoint of Siegel's circle.