Limit theorems for additive functionals of continuous time random walks

Kondratiev Y, Mishura Y, Shevchenko G. Limit theorems for additive functionals of continuous time random walks. Proceedings of the Royal Society of Edinburgh. Section A: Mathematics. 2021;151(2):799-820.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'