Limit theorems for additive functionals of continuous time random walks

For a continuous-time random walk $X=\{X_t,t\ge 0\}$ (in general non-Markov), we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X_s)ds$, $t\ge 0$. Similarly to the Markov situation, assuming that the distribution of jumps of $X$ belongs to the domain of attraction to $\alpha$-stable law with $\alpha>1$, we establish the convergence to the local time at zero of an $\alpha$-stable L\'evy motion. We further study a situation where $X$ is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda(x,\gamma)= e^{-\sum_{y\in \gamma} \phi(x-y)},$ where $\phi\colon\mathbb R\to [0,\infty)$ is a bounded function decaying sufficiently fast, and $\gamma$ is a homogeneous Poisson point process, independent of $X$. We find that in this case the weak limit has both "quenched" component depending on $\Lambda$, and a component, where $\Lambda$ is "averaged".