Laws of the iterated logarithm for occupation times of Markov processes

Abstract

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 $\Phi$ in our truncated occupation times $r\mapsto {\int }_0^{\Phi (x,r)}{1}_{B(x,r)}\left(Y_s\right)ds$ is spatially dependent on the variable $x$. Such function $\Phi (x,r)$ 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 $\Phi (x,r)$ 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 $r\mapsto {\int }_0^{\infty }{1}_{B(x,r)}\left(Y_s\right)ds$ when the process is transient. Our second main result addresses large time behaviors of occupation times $t\mapsto {\int }_0^t{1}_A\left(Y_s\right)ds$ 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.