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Parabolic problems for the Anderson model. II: Second-order asymptotics and structure of high peaks. (English) Zbl 0909.60040
Summary: This is a continuation of our previous work [Commun. Math. Phys. 132, No. 3, 613-655 (1990; Zbl 0711.60055)] on the investigation of intermittency for the parabolic equation \((\partial/ \partial t)u={\mathcal H}u\) on \(\mathbb{R}_+ \times\mathbb{Z}^d\) associated with the Anderson Hamiltonian \({\mathcal H}= \kappa\Delta +\xi(\cdot)\) for i.i.d. random potentials \(\xi (\cdot)\). For the Cauchy problem with nonnegative homogeneous initial condition we study the second order asymptotics of the statistical moments \(\langle u(t,0)^p \rangle\) and the almost sure growth of \(u(t,0)\) as \(t\to\infty\). We point out the crucial role of double exponential tails of \(\xi(0)\) for the formation of high intermittent peaks of the solution \(u(t,\cdot)\) with asymptotically finite size. The challenging motivation is to achieve a better understanding of the geometric structure of such high exceedances which in one or another sense provide the essential contribution to the solution.

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60F10 Large deviations
60K40 Other physical applications of random processes
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