The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, evaluate $u$ at the endpoint of the "reflected geodesic" with starting vector $\xi^\sharp$ at time $t$, i.e. follow the geodesic until you reach time $t$ or until you hit the boundary; if you hit a boundary, reflect and continue.
The question is: What is the infinitesimal generator of this operator semigroup? It is pretty clear that should be the usual vector field generating the geodesic flow, but how can we characterize the domain?
This should be somewhat well-known...
