  Thermal escape out of a metastable well is considered in the weak friction
regime, where the bottleneck for decay is energy diffusion, and at lower
temperatures, where quantum tunneling becomes relevant. Within a systematic
semiclassical formalism an extension of the classical diffusion equation is
derived starting from a quantum mechanical master equation. In contrast to
previous approaches finite barrier transmission also affects transition
probabilities. The decay rate is obtained from the stationary non-equilibrium
solution and captures the intimate interplay between thermal and quantum
fluctuations above the crossover to the deep quantum regime.
