Limit theorems for additive functionals of continuous time random walks

Abstract

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 $α$-stable law with $α>1$, we establish the convergence to the local time at zero of an $α$-stable Lévy motion. We further study a situation where $X$ is delayed by a random environment given by the Poisson shot-noise potential: $Λ(x,γ)= e^{-\sum_{y\in γ} ϕ(x-y)},$ where $ϕ\colon\mathbb R\to [0,\infty)$ is a bounded function decaying sufficiently fast, and $γ$ is a homogeneous Poisson point process, independent of $X$. We find that in this case the weak limit has both "quenched" component depending on $Λ$, and a component, where $Λ$ is "averaged".