A tree-like fractal Dirichlet space lying between strong and weak elliptic Harnack inequalities

TL;DR

This paper constructs a self-similar fractal tree with a Dirichlet form, demonstrating anomalous mean exit times and the validity of weak but not strong elliptic Harnack inequalities.

Contribution

It introduces a novel fractal structure and analyzes its elliptic Harnack inequalities, highlighting differences between weak and strong forms.

Findings

Mean exit time exhibits anomalous behavior.

Weak elliptic Harnack inequality holds.

Strong elliptic Harnack inequality fails.

Abstract

In this paper we construct a self-similar fractal configured as an infinitely branched tree and equip it with a regular self-similar Dirichlet form. We show anomalous behaviour of the mean exit time with respect to typical metric balls. Under properly selected self-similar measure, we further show the weak elliptic Harnack inequality holds but the strong analogue fails.