Random mass flow through random potential

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Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson Hamiltonian and its phenomenology has become a paradigmatic example for the relevance of effects resulting from a random background. A popular model for the random motion governed by this operator is the parabolic Anderson model (PAM), the discrete heat equation with random sources and sinks modeled by the Anderson Hamiltonian. A characteristic property of its solutions is the occurrence of intermittency peaks in the large time limit. They determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. In more probabilistic language, the random mass flow through the random potential concentrates on small islands, which are separated far from each other. Since the beginning of the 1990s, a lot of mathematical research has been carried out on the PAM. For some very special potential distributions, the intermittency picture and many detailed properties of the islands have been proved in recent years. However, the analysis of the time-evolution of the mass flow described by the PAM is still in its infancy. Furthermore, the knowledge about the PAM gained in recent years has not yet been applied to answer ambitious questions about more complex models, like the PAM with random diffusion rates or with the diffusivity replaced by a stable motion or by a non-symmetric random walk. Also the understanding of the influence of weak disorder (equivalently, acceleration of the diffusivity), both for the PAM and for the principal eigenvalue of the Anderson Hamiltonian, is so far restricted to some very special cases. In the project, all these questions are investigated, and the general principles underlying the effects, known in special cases, are brought to the surface. First, we prove that the mass of the PAM asymptotically concentrates on just one island, and ageing and metastability properties of the location of this island and other detailed questions about the time-evolution of the random mass flow are studied. Second, we investigate the influence of weak disorder on the PAM and on the principal eigenvalue in broad generality. Finally, we study the PAM after replacing the diffusive part of the Anderson Hamiltonian, the Laplace operator, by the generator of a random walk among random conductances or a stable random walk or a non-symmetric random walk.