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 both near zero and near infinity under some minimal assumptions. We first establish LILs of (truncated) occupation times on balls $B(x,r)$ of radii $r$ up to an function $\Phi (r)$, which is an iterated logarithm of mean exit time of $Y$, by showing that the function $\Phi$ is optimal. Our first result on LILs of occupation times covers both near zero and near infinity regardless of transience and recurrence of the process. Our assumptions are truly local in particular at zero and the function $\Phi$ in our truncated occupation times $r \mapsto\int_0^{ \Phi (x,r)} {\bf 1}_{B(x,r)}(Y_s)ds$ depends on space variable $x$ too. We also prove that a similar LIL for total occupation times $r \mapsto\int_0^\infty {\bf 1}_{B(x,r)}(Y_s)ds$ holds when the process is transient. Then we establish LIL concerning large time behaviors of occupation times $t \mapsto \int_0^t {\bf 1}_{A}(Y_s)ds$ 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.