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We focus on the problem of an impurity-free billiard with a randomposition-dependent boundary coupling to the environment. The response functionsof such an open system can be obtained non-perturbatively from a supersymmetricgenerating functional. The derivation of this functional is based on averagingover the escape rates and results in a non-linear ballistic $\sigma $-model,characterized by system-specific parameters. Particular emphasis is placed onthe {}``whispering gallery modes'' as the origin of surface diffusion modes inthe limit of large dimensionless conductance.Comment: 12 pages, no figure

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