Parabolic problems for the Anderson model. I: Intermittency and related topics.

*(English)*Zbl 0711.60055Summary: The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp “peaks” which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem
\[
(\partial /\partial t)u(t,x)=Hu(t,x),\quad u(0,x)=u_ 0(x)\geq 0,\quad (t,x)\in {\mathbb{R}}_+\times {\mathbb{Z}}^ d,
\]
for the Anderson Hamiltonian \(H=\kappa \Delta +\xi (\cdot)\), where \(\xi\) (x), \(x\in {\mathbb{Z}}^ d\), is a (generally unbounded) spatially homogeneous random potential.

This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fields u(t,\(\cdot)\) as \(t\to \infty\) are found in spectral terms of H. Rough asymptotic formulas for the statistical moments and the almost sure behavior of u(t,x) as \(t\to \infty\) are also derived.

This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fields u(t,\(\cdot)\) as \(t\to \infty\) are found in spectral terms of H. Rough asymptotic formulas for the statistical moments and the almost sure behavior of u(t,x) as \(t\to \infty\) are also derived.

##### MSC:

60H25 | Random operators and equations (aspects of stochastic analysis) |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

##### Keywords:

intermittency; hydrodynamics; Cauchy problem; Anderson Hamiltonian; percolation arguments; existence and uniqueness
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\textit{J. Gärtner} and \textit{S. A. Molchanov}, Commun. Math. Phys. 132, No. 3, 613--655 (1990; Zbl 0711.60055)

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##### References:

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