Martingale and analytic dimensions coincide under Gaussian heat kernel bounds

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.