Ballistic dynamics of a convex smooth-wall billiard with finite escape rate along the boundary

We focus on the problem of an impurity-free billiard with a random position-dependent boundary coupling to the environment. The response functions of such an open system can be obtained non-perturbatively from a supersymmetric generating functional. The derivation of this functional is based on averaging over the escape rates and results in a non-linear ballistic $\sigma $-model, characterized by system-specific parameters. Particular emphasis is placed on the {}``whispering gallery modes'' as the origin of surface diffusion modes in the limit of large dimensionless conductance.