We investigate slow propagation of light in warm vapor cells. We observe group velocities seven orders of magnitude smaller than the speed of light in vacuum. Such slow propagation allows us to measure diffusion of coherent fields revealing topological stability under diffusion,
self-similar diffusion modes, elimination of diffusion and diffraction, purification of multi-mode beams by diffusion, study of diffusion of coherence function and huge amplification of optical drag. We use this system of slow and stored light for precision spectroscopy and for sensing of inertial forces. We show that the for one and two spatial dimensions the probability of a diffusing particle to return to origin in zero (“Polya theorem”) yielding time-correlation functions with diverging moments, leading in turn to fat-tail spectra.