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Winter 2010 On the association and central limit theorem for solutions of the parabolic Anderson model
M. Cranston, G. Mueller
Illinois J. Math. 54(4): 1313-1328 (Winter 2010). DOI: 10.1215/ijm/1348505530

Abstract

We consider large scale behavior of the solution set $\{u(t,x): x\in{\mathbf{Z^d}}\}$ of the parabolic Anderson equation \begin{eqnarray*} u(t,x)&=&1+\kappa\int_0^t\Delta u(s,x)\, ds \\ &&{}+\int_0^tu(s,x)\,\partial W_x(s),\quad x\in{\mathbf{Z^d}},t\ge0, \end{eqnarray*} where $\{W_x : x\in{\mathbf{Z^d}}\}$ is a field of i.i.d. standard, one-dimensional Brownian motions, $\Delta$ is the discrete Laplacian and $\kappa>0.$ We establish that the properly normalized sum, $\sum_{x\in\Lambda _L}u(t,x),$ over spatially growing boxes $\Lambda_L=\{x\in{\mathbf {Z^d}}:\Vert x\Vert<L\}$ has an asymptotically normal distribution if the box $\Lambda_L$ grows sufficiently quickly with $t$ and provided $\kappa$ is sufficiently small depending on dimension. The asymptotic distribution of properly normalized sums over spatially growing disjoint boxes $\Lambda^1_L,\Lambda^2_L$ is asymptotically independent. Thus, on sufficiently large scales the field of solutions averaged over disjoint large boxes looks like an i.i.d. Gaussian field. We identify the variance of this Gaussian distribution in terms of the eigenfunction of the positive eigenvalue of the operator $2\kappa\Delta+\delta_0$.

Citation

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M. Cranston. G. Mueller. "On the association and central limit theorem for solutions of the parabolic Anderson model." Illinois J. Math. 54 (4) 1313 - 1328, Winter 2010. https://doi.org/10.1215/ijm/1348505530

Information

Published: Winter 2010
First available in Project Euclid: 24 September 2012

zbMATH: 1260.60117
MathSciNet: MR2981849
Digital Object Identifier: 10.1215/ijm/1348505530

Subjects:
Primary: 60F05, 60H15
Secondary: 60G60

Rights: Copyright © 2010 University of Illinois at Urbana-Champaign

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Vol.54 • No. 4 • Winter 2010
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