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Martingale and analytic dimensions

  • Sylvester Eriksson-Bique
  • Mathav Murugan
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Abstract

These notes introduce the martingale dimension of a strongly local regular Dirichlet form and the analytic dimension of a Lipschitz differentiability space. We explain why these dimensions coincide under two-sided Gaussian heat kernel estimates and discuss their relationships with other notions of dimension. We also introduce the energy image density property, its recent resolution, and its applications to martingale dimension. Finally, we show that spaces carrying diffusions with two-sided sub-Gaussian heat kernel estimates of walk dimension greater than two are not Lipschitz differentiability spaces.

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Cite

BibTeXRIS
BibTeX
@misc{erikssonbique2026martingale,
  title = {{Martingale and analytic dimensions}},
  author = {Sylvester Eriksson-Bique and Mathav Murugan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38583},
  eprint = {2609.38583},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38583},
}
RIS
TY  - UNPB
TI  - Martingale and analytic dimensions
AU  - Sylvester Eriksson-Bique
AU  - Mathav Murugan
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38583
UR  - https://arxiv.org/abs/2609.38583
AB  - These notes introduce the martingale dimension of a strongly local regular Dirichlet form and the analytic dimension of a Lipschitz differentiability space. We explain why these dimensions coincide under two-sided Gaussian heat kernel estimates and discuss their relationships with other notions of dimension. We also introduce the energy image density property, its recent resolution, and its applications to martingale dimension. Finally, we show that spaces carrying diffusions with two-sided sub-Gaussian heat kernel estimates of walk dimension greater than two are not Lipschitz differentiability spaces.
ER  -