Functional Limit Theorems for the range of stable random walks

Abstract: In this paper we establish Functional Limit Theorems for the range of random walks in $\mathbb{Z}<sup>d$ that are in the domain of attraction of a non-degenerate $\beta$-stable process in the weakly transient and recurrent regimes. These results complement the fluctuations obtained at fixed time and the functional limit Theorems obtained in the strongly transient regime. The techniques involve original ideas of Le Gall and Rosen for fluctuations and allow to show tightness in some H\"older space, thus also providing sharp regularity results about the limiting processes. The original motivation of this work is the description of functionals appearing in spatial ecology for consumption of resources induced by random motion. We apply our result to estimate the large fluctuations of energy and mortality for a simple prey predator model.