Self-Similar Markov Processes on Cantor Set

TL;DR

This paper constructs and analyzes self-similar Markov processes on the Cantor set, extending classical Brownian motion concepts to fractal spaces with detailed properties like mixing and moments.

Contribution

It introduces a new class of symmetric self-similar Markov processes on the Cantor set, defining their semigroups and analyzing their key properties.

Findings

Defined analogues of Brownian motion on the Cantor set

Described the semigroup and symmetry properties of these processes

Studied mixing behavior and moment asymptotics

Abstract

We define analogues of Brownian motion on the triadic Cantor set by introducing a few natural requirements on the Markov semigroup. We give a detailed description of these symmetric self-similar processes and study their properties such as mixing and moment asymptotics.