General Law of iterated logarithm for Markov processes: Liminf laws

TL;DR

This paper extends the understanding of liminf laws of iterated logarithm for continuous-time Markov processes, providing more general, local criteria applicable to highly inhomogeneous spaces and various jump processes.

Contribution

It introduces weaker, local assumptions for liminf LIL, covering new classes of jump and Feller processes, and broadens the applicability of these laws in complex metric measure spaces.

Findings

Established liminf LIL at zero and infinity under minimal assumptions.

Covered highly space-inhomogeneous cases including random conductance models.

Extended results to jump processes with logarithmic tail measures and variable order Feller processes.

Abstract

Continuing from arXiv:2102.01917v2, 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 arXiv:2102.01917v2, 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 arXiv:2102.01917v2 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.