Intermittency phenomena in random media

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We investigate the effect of intermittency in two models of spatially distributed random systems, the parabolic Anderson model (PAM) and the Mullins-Sekerka evolution (MSB) for Ostwald ripening. Intermittency manifests itself in a highly irregular evolution of the system determined by the appearance of more and more sparsely distributed regions of extreme behaviour. We develop new spectral methods to study the geometry of the solution to the PAM in extreme regions and to treat subdiffusivity and aging for random walks in highly irregular random environment. For the PAM, we also study refined dynamic questions like the hopping transport and deal with intermittency for time-dependent and drifted random potentials. For the MSE, we propose to construct a version of the model on the entire space and study it with particular consideration of the collisions of the randomly located particles, the screening effect, and asymptotic self-similarity of the system.