EN
Consider an initial state lying on a primary resonance island. The state may tunnel into the chaotic sea surrounding it and further escape to infinity via chaotic diffusion. Properties of transport in such a situation are studied on an exemplary system - the hydrogen atom driven by microwaves. We show that the combination of tunneling followed by chaotic diffusion leads to peculiar large scale fluctuations of the AC Stark shift and ionization rates. An appropriate random matrix model describes accurately these statistical properties.