Martingale dimensions for a class of metric measure spaces

TL;DR

The paper develops a purely analytic framework to determine the AF-martingale dimension $d_{ m m}$ for diffusions defined by strongly local regular Dirichlet forms on metric measure spaces. Central to the approach is a local energy-measure and relative-capacity balance, paired with a novel blow-up and push-forward scheme for harmonic functions, which avoids reliance on heat kernel bounds or global self-similarity. Under the proposed assumptions, the AF-martingale dimension collapses to one, indicating that the intrinsic stochastic structure remains effectively one-dimensional even on highly inhomogeneous fractal spaces, such as inhomogeneous Sierpinski gaskets. The results encompass and extend previous self-similarity-based findings, offering analytic criteria for one-dimensional probabilistic structure across broad classes of fractal-like spaces and suggesting wide applicability in analysis on metric measure spaces.

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.

Figures (4)

• Figure 1: Hierarchical partitions used in (A2). Left: Coarse partition $\{U_k^{(1)}\}$ with inner subsets $\{V_k^{(1)}\}$. Right: A finer subdivision $\{U_k^{(2)}\}$ with inner subsets $\{V_k^{(2)}\}$. Although the sets appear similar, no geometric self-similarity is actually assumed.
• Figure 2: (Adapted from Hi10.) Examples of p.c.f. self-similar sets. From the upper left, two- and three-dimensional standard Sierpinski gasket, Pentakun (pentagasket), snowflake, the Vicsek set, and Hata's tree-like set.
• Figure 3: (Quoted from HY22) Illustration of $K_i^{(l)}$ ($i=1,2,\dots,N(l)$) when $d=2$ and $l=2,3,4$, respectively. Pay attention to the choice of $K_1^{(l)}$, $K_2^{(l)}$, and $K_3^{(l)}$.
• Figure 4: (Quoted from Hi25.) An example of inhomogeneous Sierpinski gaskets with $d=2$ and $T=\{2,3\}$ (the upper figure). Here, $L=\{L_w\}_{w\in W_*}$ is given by $L_\emptyset=3$, $L_{1^3}=L_{2^3}=L_{5^3}=2$, $L_{3^3}=L_{4^3}=L_{6^3}=3$, $L_{1^3 1^2}=L_{1^3 2^2}=L_{1^3 3^2}=2$, $L_{2^3 1^2}=2$, $L_{2^3 2^2}=L_{2^3 3^2}=3$, $L_{3^3 1^3}=L_{3^3 3^3}=L_{3^3 4^3}=L_{3^3 5^3}=L_{3^3 6^3}=2$, $L_{3^3 2^3}=3$, etc. The indices are indicated in the middle and lower figures.

Theorems & Definitions (51)

• Definition 2.1: Hi10
• Proposition 2.2: Hi10, see also Na85
• Definition 2.3: Hi10
• Theorem 2.4: Hi10
• Lemma 3.1: FOT11
• Theorem 3.3
• Remark 3.4
• Lemma 3.5
• proof
• Lemma 3.6