Regular and singular continuous time random walk in dynamical random environment

We consider a homogeneous continuous-time random walk (CTRW) on the lattice $Z^{d}$, $d=1,2,ldots$ 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 $dgeq 1$. 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.