Measurable Riemannian structures associated with strong local Dirichlet forms

TL;DR

This paper develops Riemannian-like structures linked to strong local Dirichlet forms on general spaces, providing a geometric interpretation of the Dirichlet form's index as an effective local dimension, with applications to stochastic analysis.

Contribution

It introduces a new geometric framework connecting Dirichlet forms with Riemannian structures, enhancing understanding of local dimensions and differentiations in stochastic analysis.

Findings

Pointwise index of Dirichlet form represents local tangent space dimension.

Established a notion of differentiations of functions in this framework.

Applied the structures to problems in stochastic analysis.

Abstract

We introduce Riemannian-like structures associated with strong local Dirichlet forms on general state spaces. Such structures justify the principle that the pointwise index of the Dirichlet form represents the effective dimension of the virtual tangent space at each point. The concept of differentiations of functions is studied, and an application to stochastic analysis is presented.