Radio waveguides and diffusion processes on differentiable manifolds

The paper studies in detail a problem related to the geometry of the group of transmission matrices of a single-wave waveguide, namely, the possible appearance in the lossless case of Brownian motion on the Lobachevsky plane describing the behavior of the reflection coefficient. It is shown that within the framework of the models of Gertsenshtein and Vasil'ev (1959) and Tutubalin (1971), Brownian motion cannot arise under any constraints imposed on the probability distribution. After subtracting the phase advance, a diffusion process is, indeed, obtained, but the differential operator corresponding to it is the sum of the Beltrami-Laplace operator for the Lobachevsky plane and a second-order differential operator with constant coefficients, not all of which can be zero simultaneously. The resulting probability distribution is therefore invariant with respect to rotations of the Lobachevsky plane around the point with Beltrami coordinates (0,0). Some results are also obtained for a lossy waveguide.