Martingale and analytic dimensions coincide under Gaussian heat kernel bounds

TL;DR

This paper proves that in metric measure spaces with Gaussian heat kernel bounds, the martingale dimension of the associated diffusion process matches Cheeger's analytic dimension, linking probabilistic and geometric structures.

Contribution

It establishes the equality of martingale and analytic dimensions under Gaussian heat kernel bounds, extending previous bounds to more general spaces.

Findings

Martingale dimension equals Cheeger's analytic dimension almost everywhere.

Martingale dimension is bounded above by the Hausdorff dimension.

Extension of previous bounds to broader classes of spaces.

Abstract

Given a strongly local Dirichlet form on a metric measure space that satisfies Gaussian heat kernel bounds, we show that the martingale dimension of the associated diffusion process coincides with Cheeger's analytic dimension of the underlying metric measure space. More precisely, we show that the pointwise version of the martingale dimension introduced by Hino (called the pointwise index) almost everywhere equals the pointwise dimension of the measurable differentiable structure constructed by Cheeger. Using known properties of spaces that admit a measurable differentiable structure, we show that the martingale dimension is bounded from above by the Hausdorff dimension of the underlying metric space, thereby extending an earlier bound obtained by Hino for some self-similar sets.