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Geometric characterization of intermittency in the parabolic Anderson model. (English) Zbl 1126.60091

The authors study the parabolic Anderson problem with localised initial datum, that is the Cauchy problem \[ \partial_t u(t,x)=\Delta u(t,x) + \xi(x) u(t,x), \qquad (t,x)\in (0,\infty)\times \mathbb Z^d, \] and \( u(0,x)=\delta_0(x)\) for all \(x\in \mathbb Z^d\). Here \(\xi=(\xi(x))_{x\in \mathbb Z^d}\) is a random i.i.d. potential. It is assumed that \(\xi(x)\) have double-exponential (\(P[\xi(x) >r] \sim \exp\{-e^{r/\rho }\}\)) or heavier tails. The main goal of the paper is to provide a mathematical foundation for the usual geometric description of intermittency: “With probability one, as \(t\to\infty\), the random field \(u(t,x)\) develops high peaks on islands which are located far from each other and which give the overwhelming contribution to the total mass \(U(t)=\sum_{x\in \mathbb Z^d}u(t,x)\).” This description was never proved rigorously prior to this paper, even if the intermittency has been studied intensively. It was observed only by indirect means by comparing various asymptotics of \(U(t)\) and its moments [see J. Gärtner and S. A. Molchanov, Probab. Theory Relat. Fields 111, No. 1, 17–55 (1998; Zbl 0909.60040) or R. van der Hofstad, W. König and P. Mörters, Commun. Math. Phys. 267, No. 2, 307–353 (2006; Zbl 1115.82030)]. These asymptotics were described in terms of a certain variational problem. In the present paper it is assumed that this variational problem possesses, up to spatial shifts, a unique minimizer \(V\) with a unique maximum. Then, with probability one, there exists a random \(t\)-dependent subset \(\Gamma^\star\) of the centred cube \(B_{t\log^2 t}\) with side length \(2t\log^2 t\), such that (i) an arbitrarily large fraction of the total mass \(U(t)\) is contained in a fixed-size neighbourhood of \(\Gamma^\star\) as \(t\to\infty\), (ii) \(| \Gamma^\star| < t^{o(1)}\) and the minimal distance of two points in \(\Gamma^\star\) increases as \(t^{1-o(1)}\), (iii) the asymptotic shape of the potential \(\xi\) in an arbitrary fixed-size neighbourhood of \(\Gamma^\star\) can be described in terms of the deterministic minimizer \(V\), (iv) the asymptotic shape of the properly rescaled solution \(u(t,x)\) in an arbitrary fixed-size neighbourhood of \(\Gamma^\star\) is given by the unique non-negative eigenfunction \(w\) (normalised to \(w(0)=1\)) of the operator \(\Delta + V\) on \(\ell^2(\mathbb Z^d)\). The assumption on the existence of the unique minimizer is further verified if the constant \(\rho \) determining the tail of the distribution of \(\xi\) is large enough.

MSC:

60K40 Other physical applications of random processes
60H25 Random operators and equations (aspects of stochastic analysis)
35B40 Asymptotic behavior of solutions to PDEs
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60F10 Large deviations
47B80 Random linear operators
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References:

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