Limit theorems for additive functionals of continuous time random walks

For a continuous-time random walk X = {X-t, t >= 0} (in general non-Markov), we study the asymptotic behaviour, as t -> infinity, of the normalized additive functional c(t) integral(t)(0) f(X.)ds, t >= 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 Levy 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(-Sigma y is an element of gamma phi(x-y)), where phi: R -> [0, infinity) 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'.