Laws of the iterated logarithm for occupation times of Markov processes

In this paper, we discuss the laws of the iterated logarithm (LIL) for occupation times of Markov processes Y in general metric measure space near zero (near infinity, respectively) under minimal assumptions around zero (near infinity, respectively). The LILs near zero in this paper cover the case that the function Φ in our truncated occupation times r↦∫0Φ(x,r)1B(x,r)(Ys)ds is spatially dependent on the variable x. Such function Φ(x,r) is an iterated logarithm of mean exit times of Y from balls B(x,r) of radius r. We first establish LILs of (truncated) occupation times on balls B(x,r) up to the function Φ(x,r) Our first result on LILs of occupation times covers both near zero and near infinity cases, irrespective of transience and recurrence of the process. Further, we establish a similar LIL for total occupation times r↦∫0∞1B(x,r)(Ys)ds when the process is transient. Our second main result addresses large time behaviors of occupation times t↦∫0t1A(Ys)ds under an additional condition that guarantees the recurrence of the process. Our results cover a large class of Feller (Levy-like) processes, random conductance models with long range jumps, jump processes with mixed polynomial local growths and jump processes with singular jumping kernels.