Martingale dimensions for a class of metric measure spaces

TL;DR

This paper develops an analytic framework to determine the AF-martingale dimension of diffusion processes on metric measure spaces, showing it collapses to one under certain conditions, even on highly irregular spaces.

Contribution

It introduces a new purely analytic approach based on energy measures and capacities, avoiding reliance on self-similarity or geometric models.

Findings

AF-martingale dimension is one under localized analytic conditions.

The method applies to inhomogeneous fractals like Sierpinski gaskets.

The approach controls behavior across scales without heat kernel bounds.

Abstract

We establish a general analytic framework for determining the AF-martingale dimension of diffusion processes associated with strongly local regular Dirichlet forms on metric measure spaces. While previous approaches typically relied on self-similarity, our argument is based instead on purely analytic balance conditions between energy measures and relative capacities. Under this localized analytic condition, we prove that the AF-martingale dimension collapses to one, thereby indicating that the intrinsic stochastic structure remains effectively one-dimensional even on highly irregular or inhomogeneous spaces. As a key technical ingredient, our proof employs a simultaneous blow-up and push-forward scheme for harmonic functions and their energy measures, allowing us to control the limiting behavior across scales without invoking heat kernel bounds or explicit geometric models. The main theorem is applied in particular to inhomogeneous Sierpinski gaskets, which do not possess self-similarity or uniform geometric structure. Our method provides a general analytic perspective that can be used to study the one-dimensional probabilistic structure of diffusions through martingale additive functionals.