Correlation structure of intermittency in the parabolic Anderson model
This paper is concerned with the parabolic Anderson model, which corresponds to the Cauchy problem with the Anderson Hamiltonian \(\kappa\Delta+ \xi\), \(\Delta\) being the discrete Laplacian, \(\kappa\in(0,\infty)\) a diffusion constant and \(\xi=[\xi(x)\), \(x\in\mathbb{R}^d]\) an i.i.d. random field taking values in \(\mathbb{R}\). Depending on \(\kappa\) and the marginal law for \(\xi\), the model can represent various physical and chemical properties. In a paper of \textit{J. Gärtner} and \textit{S. A. Molchanov} [Commun. Math. Phys. 132, No. 3, 613-655 (1990; Zbl 0711.60055)], it is shown that if \(\xi\) is not constant, the system is intermittent. That is, qualitatively, the solution field develops sparsely distributed high peaks as \(t\) increases. Here it is shown that there is a qualitative change in the structure of the intermittent peaks when the law of \(\xi(0)\) is in the vicinity of the double exponential law: \(P(\xi(\theta) >s)= \exp (-e^{-s/ \theta})\). Indeed, this law is the only one producing a nondegenerate correlation system for the solution.
scientific article; zbMATH DE number 1309029