General Law of iterated logarithm for Markov processes: Liminf laws

Abstract

Continuing from Cho, Kim, and Lee [General Law of iterated logarithm for Markov processes: Limsup law, arXiv:2102,01917v3], in this paper, we discuss general criteria and forms of liminf laws of iterated logarithm (LIL) for continuous-time Markov processes. Under some minimal assumptions, which are weaker than those in Cho et al., we establish liminf LIL at zero (at infinity, respectively) in general metric measure spaces. In particular, our assumptions for liminf law of LIL at zero and the form of liminf LIL are truly local so that we can cover highly space-inhomogenous cases. Our results cover all examples in Cho et al. including random conductance models with long range jumps. Moreover, we show that the general form of liminf law of LIL at zero holds for a large class of jump processes whose jumping measures have logarithmic tails and Feller processes with symbols of varying order which are not covered before.