Regular and singular continuous time random walk in dynamic random environment

We consider a homogeneous continuous-time random walk (CTRW) on the lattice $\mathbb Z^{d}$, $d=1,2,\dots$, which is a kind of random trap model in a time-dependent (“dynamic”) environment. The waiting time distribution is renewed at each jump, and is given by a general probability density depending on a parameter $\eta>0$ such that the average waiting time is finite for $\eta >1$ and infinite for $\eta \in (0, 1]$. By applying analytic methods introduced in a previous paper we prove that the asymptotics of the quenched CTRW and of its annealed version are the same for all $\eta >0$ and $d\geq 1$. We also exhibit explicit formulas for the correction term to the quenched asymptotics. For the border-line case $\eta=1$ we find an explicit expression for the annealed limiting distribution, which is, to our knowledge, new.