NDLI: Quenched to annealed transition in the parabolic Anderson problem

Content Provider	Springer Nature Link
Author	Molchav, S. Cranston, M.
Copyright Year	2006
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 $$\Lambda_L=\left\{x\in {\mathbf{Z^d}}:\|x\|\le L\right\}$$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ 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 $$(0,\eta(1))$$
Ending Page	193
Page Count	17
Starting Page	177
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	1
Volume Number	138
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-08-08
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical and Computational Physics Parabolic Anderson model Probability Theory and Stochastic Processes Large deviations Infinitely divisible distributions; stable distributions Quantitative Finance Annealed asymptotics Quenched asymptotics Central limit theorem Extreme value theory; extremal processes Statistics for Business/Economics/Mathematical Finance/Insurance Operations Research/Decision Theory Law of large numbers Mathematical Biology in General Central limit and other weak theorems
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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