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By Turner J., Kautz W.H.

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Therefore, the measure with density φ already defined can be normalized to define a probability measure. Similarly, one can prove part 4. The proof of part 1 in the case E[( log ρ)2 ] > 0 is analogous to the proof of the recurrent case of Theorem 1. The case E[( log ρ)2 ] = 0 is trivial, since in this case we would be in the situation of simple random walk. 5 Absolutely Continuous Invariant Measures and Some Implications The existence of an invariant probability measure which is absolutely continuous with respect to the initial distribution of the environment will turn out to be crucial in the study of the model.

Open Question 2 Let d ≥ 2. Assume given a RWRE for which (UE) and (IID) are fulfilled, and which is transient in direction l ∈ Sd−1 . Is the RWRE necessarily ballistic in direction l? As it is discussed above, Example 1 shows that if the hypothesis (UE) is replaced by (E) in the Open Question 2, then its answer is negative. The following proposition gives an indication of how much ellipticity should be required. d. environment. Assume that max E e∈U 1 = ∞. 1 − ω(0, e)ω(0, −e) (31) Then the walk is not ballistic in any direction.

The answer to this problem (and further interesting insight) has been obtained by Kesten, Kozlov, and Spitzer [24]: In fact, there is a direct connection between the exponent κ ∈ (0, 1) characterized by E[ρ κ ] = 1, and the typical order of Xn in this case, which is nκ . We refer the reader to [24] for further details. In addition, from the above discussion we see that in dimension d = 1, if the family of integer shifts is ergodic with respect to the law P of the environment, the walk being transient to the right or left does not ensure the existence of an invariant probability measure for the environmental process which is absolutely continuous with respect to P.

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