Random walk in a regenerating random environment

In this paper we consider random walks moving in a dynamic random environment that is completely resampled with probability $p$ after each step of the random walker. This resampling yields a straightforward regeneration structure and limit theorems, such as a law of large numbers with a limiting speed $v(p)$ that depends on $p$. We investigate several properties of $v(p)$ as a function of $p$, such as differentiability and monotonicity. Similarily, we study the limiting diffusivity ${\sigma }^2(p)$ in a central limit theorem.