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Bounding martingale dimension by spectral dimension

  • Aobo Chen
  • Mathav Murugan
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Abstract

We prove that the martingale dimension of a broad class of strongly local regular Dirichlet forms is bounded above by a small-scale version of spectral dimension. This resolves a conjecture of Hino for this class of spaces, extending his earlier results beyond self-similar fractals. Our proof uses differentiation of energy measures to select good blow-up scales and construct approximately harmonic coordinates. We then study weak limits of their Euclidean image measures and combine an integration-by-parts argument with uniform Sobolev inequalities to establish the dimension bound.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026bounding,
  title = {{Bounding martingale dimension by spectral dimension}},
  author = {Aobo Chen and Mathav Murugan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39312},
  eprint = {2610.39312},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39312},
}
RIS
TY  - UNPB
TI  - Bounding martingale dimension by spectral dimension
AU  - Aobo Chen
AU  - Mathav Murugan
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39312
UR  - https://arxiv.org/abs/2610.39312
AB  - We prove that the martingale dimension of a broad class of strongly local regular Dirichlet forms is bounded above by a small-scale version of spectral dimension. This resolves a conjecture of Hino for this class of spaces, extending his earlier results beyond self-similar fractals. Our proof uses differentiation of energy measures to select good blow-up scales and construct approximately harmonic coordinates. We then study weak limits of their Euclidean image measures and combine an integration-by-parts argument with uniform Sobolev inequalities to establish the dimension bound.
ER  -