The author considers a discrete-time random walk {X t } t=0 ∞ on ℤ ν for small dimension ν=1,2 with transition probabilities P(X t+1 =y∣X t =x,ξ)=P 0 (y-x)+εc(y-x,ξ t (x)), where ξ={ξ t (x), t=0,1,⋯, x∈ℤ ν } is a random environment such that the variables ξ t (x) are i.i.d., take values in a finite set and are such that 〈c(u,·)=0〉=0, where 〈·〉 denotes average with respect to the environment. ε is a small parameter. The author shows that if is small enough and f is a smooth function, then as T→∞ the quantity T lnT∑ x∈ℤ 2 [P(X T =x∣X 0 =0,ξ)-P 0 T (x)]fx-bT T where b is the drift of the averaged random walk P 0 , tends in distribution to a gaussian random variable with covariance given by a suitable functional of f. Further results are proved for the random correction to the mean value and the covariance in dimension ν=1,2.

Anomalous behaviour of the correction to the central limit theorem for a model of random walk in random media.

DI PERSIO, Luca
2010

Abstract

The author considers a discrete-time random walk {X t } t=0 ∞ on ℤ ν for small dimension ν=1,2 with transition probabilities P(X t+1 =y∣X t =x,ξ)=P 0 (y-x)+εc(y-x,ξ t (x)), where ξ={ξ t (x), t=0,1,⋯, x∈ℤ ν } is a random environment such that the variables ξ t (x) are i.i.d., take values in a finite set and are such that 〈c(u,·)=0〉=0, where 〈·〉 denotes average with respect to the environment. ε is a small parameter. The author shows that if is small enough and f is a smooth function, then as T→∞ the quantity T lnT∑ x∈ℤ 2 [P(X T =x∣X 0 =0,ξ)-P 0 T (x)]fx-bT T where b is the drift of the averaged random walk P 0 , tends in distribution to a gaussian random variable with covariance given by a suitable functional of f. Further results are proved for the random correction to the mean value and the covariance in dimension ν=1,2.
random walk in random media; asymptotics; central limit theorem; Processes in random environments; Sums of independent random variables
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11562/744767
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