Laws of the iterated logarithm for occupation times of Markov processes

TL;DR

This paper establishes laws of the iterated logarithm for occupation times of Markov processes in metric measure spaces, covering near-zero, near-infinity, and large-time behaviors under minimal assumptions.

Contribution

It introduces new LIL results for occupation times that are optimal, local, and applicable to a broad class of Markov processes, including transient and recurrent cases.

Findings

LILs for occupation times near zero and infinity are established.

Optimality of the function (r) related to mean exit times is shown.

Large-time occupation time behaviors are characterized under recurrence conditions.

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.