Asymptotic properties of the parabolic Anderson model

Abstract

The parabolic Anderson model is the heat equation on the lattice with a random potential. A characteristic feature of the large time asymptotics of the solution is the occurrence of small islands where almost all mass is concentrated. This effect is called intermittency. There are basically two ways of looking at the long time behaviour of the solution. On the one hand one can consider the almost sure asymptotics after one realisation of the potential is fixed (the quenched setting). On the other hand one can consider he asymptotics after taking expectation with respect to the potential (the annealed setting). One possible characterisation of intermittency is to compare the asymptotics of different moments of the solution. We derive asymptotic formulas for time correlations of regularly varying functions of the solution in the case of a homogeneous initial condition and an appropriate time-independent potential. These formulas can be used, for instance, to calculate moments of the solution of all orders up to asymptotic equivalence. Furthermore, we show what the geometry of the intermittency peaks that determine the annealed behaviour looks like. More precisely, we show what the height, the size and the frequency of the relevant peaks are. We also investigate ageing properties of the model under different definitions. In particular, we examine for how long intermittency peaks remain relevant. Another characterisation of intermittency is to compare the quenched asymptotics of the solution with the annealed asymptotics. If one considers the averaged solution to the parabolic Anderson model with homogeneous initial condition in a growing box that is time-dependent, then one observes different regimes. If the growth rate of the box is small, then one observes quenched behaviour, whereas if the box grows fast one observes annealed behaviour. We derive stable limit theorems for the averaged solution depending on the growth rate of the box and the potential tails for suitable potentials to describe the transition from quenched to annealed behaviour. Furthermore, we give sufficient conditions on the growth of the box for a strong law of large numbers to hold. Finally, we derive asymptotic formulas for time and spatial correlations in the parabolic Anderson model with a (time-dependent) white-noise potential. Acknowledgement I would like to express my gratitude to Jurgen Gartner for introducing me to the topic and his invaluable support over the last years. Furthermore, I would like to thank Wolfgang Konig for his kind supervision and many helpful discussions and hints on how to present mathematical results. In addition, I would like to thank the professors in the committee for undertaking this task. Moreover, I am indepted to the probability group at TU Berlin and the IRTG SMCP for a friendly working atmosphere. I am grateful for finanical support of the DFG and the BMS. Finally, I thank my parents.