General Law of iterated logarithm for Markov processes: Limsup law

In this paper, we discuss general criteria of limsup law of iterated logarithm (LIL) for continuous-time Markov processes. We consider minimal assumptions for LILs to hold at zero(at infinity, respectively) in general metric measure spaces. We establish LILs under local assumptions near zero (near infinity, respectively) on uniform bounds of the expectations of first exit times from balls in terms of a function $\phi$ and uniform bounds on the tails of the jumping kernel in terms of a function $\psi$. The main result is that a simple ratio test in terms of the functions $\phi$ and $\psi$ completely determines whether there exists a positive non-decreasing function $\Psi$ such that $\limsup |X_t|/\Psi(t)$ is positive and finite a.s., or not. Our results cover a large class of subordinate diffusions, jump processes with mixed polynomial local growths, jump processes with singular jumping kernels and random conductance models with long range jumps.