The statistics of the trajectory of a certain billiard in a flat two-torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius epsilon > 0 from the flat torus T-2, with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity 0) only. Let (τ) over tilde (epsilon)(omega), and respectively (R) over tilde (epsilon)(omega), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when T-2 is being identified with [0, 1)(2). We prove that the probability measures on [0, infinity) associated with the random variables epsilon(τ) over tilde (epsilon) and epsilon(R) over tilde (epsilon) are weakly convergent as epsilon --> 0(+) and explicitly compute the densities of the limits.