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The Einstein Relation on Metric Measure Spaces

Fabian Burghart, Uta Freiberg

TL;DR

This work constructs a general framework for the Einstein relation on metric measure spaces by linking geometric (Hausdorff), spectral, and stochastic (walk) dimensions through Dirichlet forms and Hunt processes. It proves invariance of the spectral dimension under transported operators and bi-Lipschitz maps, and analyzes how Hölder regular transformations distort dimensions, yielding nontrivial constants in cases like graphs of fractional Brownian motion. The paper validates the relation on classical spaces (Euclidean domains) and on fractals (the Sierpiński gasket) via two constructive routes and highlights when the relation remains tight (c=1) or changes (c≠1). It further outlines a program toward a comprehensive theory, including open problems about lower bounds, dimension choices, and extensions to graphs and metric graphs with mm-structure.

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 Hölder 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.

The Einstein Relation on Metric Measure Spaces

TL;DR

This work constructs a general framework for the Einstein relation on metric measure spaces by linking geometric (Hausdorff), spectral, and stochastic (walk) dimensions through Dirichlet forms and Hunt processes. It proves invariance of the spectral dimension under transported operators and bi-Lipschitz maps, and analyzes how Hölder regular transformations distort dimensions, yielding nontrivial constants in cases like graphs of fractional Brownian motion. The paper validates the relation on classical spaces (Euclidean domains) and on fractals (the Sierpiński gasket) via two constructive routes and highlights when the relation remains tight (c=1) or changes (c≠1). It further outlines a program toward a comprehensive theory, including open problems about lower bounds, dimension choices, and extensions to graphs and metric graphs with mm-structure.

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 Hölder 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.

Paper Structure

This paper contains 23 sections, 13 theorems, 114 equations, 4 figures.

Table of Contents

  1. Fractal Dimensions and the Einstein Relation
  2. Hausdorff measure and Hausdorff dimension
  3. Weyl asymptotics and spectral dimension
  4. The classical case
  5. The general case
  6. Markov processes and walk dimension
  7. From Dirichlet forms to Markov processes
  8. Local walk dimension and Einstein relation
  9. Related works
  10. Examples and Non-examples
  11. Euclidean Space
  12. The Sierpiński Gasket
  13. Approximation by Dirichlet forms
  14. Approximation by random walks
  15. The real line as bounded metric space
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $\mathscr{S}=\{\mathcal{S}_1,\dots,\mathcal{S}_N\}$ be a finite set of strict contractions on the Euclidean space $\mathbb{R}^n$. Then there exists a unique nonempty compact set denoted by $|\mathscr{S}|$ which is invariant under $\mathscr{S}$, i.e. Furthermore, assume that $\mathscr{S}$ satisfies the open set condition (OSC), meaning that there exists a nonempty bounded open set $O\subseteq

Figures (4)

  • Figure 1: An approximation of $\operatorname{SG}$
  • Figure 2: The graphs $G_0,G_1,G_2$
  • Figure 3: The values of $u$ on $V_1$
  • Figure 4: A fractional Brownian motion, with the balls $B_\infty(\cdot,r)$ indicated for the points over $T$ (in red) and $T^\pm$ (in grey) on the graph.

Theorems & Definitions (34)

  • Definition 1.1: Hausdorff outer measure
  • Definition 1.2
  • Theorem 1.3: Hutchinson, hutchinson1981fractals
  • Lemma 1.4: Mass distribution principle, Frostmann, mattila1999geometry
  • Definition 1.6
  • Theorem 1.7: Hille-Yosida,ma2012introduction
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • Definition 1.11
  • ...and 24 more