  We propose a model of a one-dimensional random walk in dynamic random
environment that interpolates between two classical settings: (I) the random
environment is sampled at time zero only; (II) the random environment is
resampled at every unit of time. In our model the random environment is
resampled along an increasing sequence of deterministic times. We consider the
annealed version of the model, and look at three growth regimes for the
resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove weak
laws of large numbers and central limit theorems. We list some open problems
and conjecture the presence of a crossover for the scaling behaviour in regimes
(R2) and (R3).

Probability