Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive
The authors deal with a general model of random walk \(X_t\) on the \(\gamma\)-dimensional lattice \(\mathbb Z^\gamma\) in a random environment \(\xi=\{\xi(t,x): (t,x)\in \mathbb Z^{\gamma+1}\}\) with i.i.d. components taking values in some space \(S\) and distributed according to a non-degenerate probability measure \(\pi\). The random walk transition probabilities satisfy \[ P(X_{t+1} = y \mid X_t = x,\xi ) = P_0(y-x) + c (y-x;\xi) \] where \(\{P_0(u): u\in \mathbb Z^\gamma\}\) are the transition probabilities of a homogeneous completely irreducible random walk with the restriction that for all \(u \in \mathbb Z^\gamma\), \(E[c(u;\;\;)]=0\). It is further assumed that \(P_0\) and \(c\) go to zero at \(\infty\): \[ \sum_{u\in Z^\gamma} \Bigl\{P_0(u)+\sup_s |c(u,s)|\Bigr\}e^{d|u|}<\infty. \] Under some further conditions, the authors establish the central limit theorem and also show that the environment does not affect the limit distribution.
scientific article; zbMATH DE number 2111539