# The Einstein Relation on Metric Measure Spaces

## TL;DR

This paper explores the Einstein relation in metric measure spaces, demonstrating its invariance under bi-Lipschitz maps and analyzing how non-Lipschitz transformations affect fractal dimensions and the relation.

## Contribution

It extends the Einstein relation to metric measure spaces with suitable operators and studies its invariance under bi-Lipschitz isomorphisms, including effects of non-Lipschitz maps.

## Key findings

- Invariance of fractal dimensions under bi-Lipschitz maps
- Distortion effects of H"older transformations on local walk dimension
- Application to graphs of fractional Brownian motions

## Abstract

This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at H\"older regular transformations and how they influence the local walk dimension and describe the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.