Abstract
We study the quenched and annealed asymptotics for the solutions of the lattice parabolic Anderson problem in the situation in which the underlying random walk has long jumps and belongs to the domain of attraction of the stable process. This type of stochastic dynamics has appeared in recent work on the evolution of populations.
The i.i.d random potential in our case is unbounded from above with regular Weibull type tails. Similar models but with the local basic Hamiltonian (lattice Laplacian) were analyzed in the very first work on intermittency for the parabolic Anderson problem by J. Gärtner and S. Molchanov.
We will show that the long-range model demonstrates the new effect. The annealed (moment) and quenched (almost sure) asymptotics of the solution have the same order in contrast to the case of the local models for which these orders are essentially different.
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Molchanov, S., Zhang, H. (2012). The Parabolic Anderson Model with Long Range Basic Hamiltonian and Weibull Type Random Potential. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_2
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