Abstract
We study limit behavior for sums of the form \(\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),\) where the field \(\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}\)is composed of solutions of the parabolic Anderson equation
The index set is a box in Z d, namely \(\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}\) and L = L(t) is a nondecreasing function \(L : [0,\infty)\rightarrow {\bf R}^{+}. \) We identify two critical parameters \(\eta(1) < \eta(2)\) such that for \(\gamma > \eta(1)\) and L(t) = eγ t, the sums \(\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\) satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For \(\gamma > \eta(2)\) and L(t) = eγ t, one has \(\sum_{x\in \Lambda_L}u(t,x)\) when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when \(\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,\) quenched asymptotics occur. That means \(\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),\) where \(\gamma(\kappa)\) is the almost sure Lyapunov exponent, i.e. \(\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).\) We also examine the behavior of \(\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\) for L = e γ t with γ in the transition range \((0,\eta(1))\)
References
Arous, B., Molchanov, R.: Transition from the annealed to the quenched asymptotics for a random walk on obstacles. Ann. Probab. (in press)
Carmona R.A., Molchanov S.A.(1994): Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518): viii+125
Cranston M., Molchanov S. (2005): Limit laws for sums of products of exponentials of iid random variables. Israel J. Math. 148, 115–136
Cranston M., Mountford T., Shiga T. (2002): Lyapunov exponents for the parabolic Anderson model acta. Math. Univ. Commun. LXXI 2, 163–188
Gärtner J., König W. (2005): Interacting stochastic systems. Springer, Berlin Heidelberg New York
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Research of first author supported by NSF grant DMS-0450756 of the second author by NSF grant DMS-0405927
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Cranston, M., Molchanov, S. Quenched to annealed transition in the parabolic Anderson problem. Probab. Theory Relat. Fields 138, 177–193 (2007). https://doi.org/10.1007/s00440-006-0020-7
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DOI: https://doi.org/10.1007/s00440-006-0020-7
Keywords
- Parabolic Anderson model
- Central limit theorem
- Law of large numbers
- Quenched asymptotics
- Annealed asymptotics
Mathematics Subject Classifications (2000)
- Primary 60F05
- Primary 60F10
- Secondary 60E07
- Secondary 60G70