Local diffusion theory of localized waves in open media

We report a first-principles study of static transport of localized waves in quasi-one-dimensional open media. We found that such transport, dominated by disorder-induced resonant transmissions, displays novel diffusive behavior. Our analytical predictions are entirely confirmed by numerical simulations. We showed that the prevailing self-consistent localization theory [van Tiggelen, {\it et. al.}, Phys. Rev. Lett. $\mathbf{\text{84}}$, 4333 (2000)] is valid only if disorder-induced resonant transmissions are negligible. Our findings open a new direction in the study of Anderson localization in open media.