Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

Abstract

We study the solutions $u=u(x,t)$ to the Cauchy problem on ${\mathbb{Z}}^d\times (0,\infty )$ for the parabolic equation ${\partial }_{t}u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on ${\mathbb{Z}}^d$ and $\xi =(\xi (z){)}_{z\in {\mathbb{Z}}^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):={\sum }_{x}u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\Delta +\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t\mapsto \frac{1}{t}\log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $\Delta +\xi$ in large sets recently proved by the first two authors.

69 pages, version to appear in Prob. Theory Rel. Fields

References in corpus (3)

• A two cities theorem for the parabolic Anderson model
• Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials
• From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian

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