Diffusivity of Stochastic Billiards in a random tube

Consider a random tube which stretches to infinity in the direction of the first coordinate, and which is stationary and ergodic, and also well-behaved in some sense. When strictly inside the tube, the particle ("ball") moves straight with constant speed. Upon hitting the boundary, it is reflected randomly: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time random walk formed by the particle's positions when hitting the boundary. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard. Joint work with S.Popov, G.Schütz and M. Vachkovskaia.