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The canonical SLE-reversible Weil--Petersson response diffusion

  • Chunhao Cai
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Abstract

The rough response calculus for the rotationally marked SLE welding law gives a closed Weil--Petersson energy but does not by itself give a process on rough weldings. For 0<κ≤4, we prove that this form is quasi-regular in the natural topology of circle homeomorphisms and therefore generates a canonical diffusion reversible with respect to the SLE welding probability. The proof constructs a compact form nest from inverse arc masses. We identify the generator as −D*D and prove absolute L2 convergence of its Fourier expansion on the response cylinder core, together with the Fukushima decomposition. A common Fourier noise and planar estimates preventing boundary collapse yield a second quasi-regular realization by normalized Jordan welding triples. The diffusion is irreducible and converges strongly in L2 to equilibrium. For 0<κ<4, the complement of the conformally removable welding locus has zero capacity.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026canonical,
  title = {{The canonical SLE-reversible Weil--Petersson response diffusion}},
  author = {Chunhao Cai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00407},
  eprint = {2610.00407},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00407},
}
RIS
TY  - UNPB
TI  - The canonical SLE-reversible Weil--Petersson response diffusion
AU  - Chunhao Cai
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00407
UR  - https://arxiv.org/abs/2610.00407
AB  - The rough response calculus for the rotationally marked SLE welding law gives a closed Weil--Petersson energy but does not by itself give a process on rough weldings. For 0<κ≤4, we prove that this form is quasi-regular in the natural topology of circle homeomorphisms and therefore generates a canonical diffusion reversible with respect to the SLE welding probability. The proof constructs a compact form nest from inverse arc masses. We identify the generator as −D^*D and prove absolute L^2 convergence of its Fourier expansion on the response cylinder core, together with the Fukushima decomposition. A common Fourier noise and planar estimates preventing boundary collapse yield a second quasi-regular realization by normalized Jordan welding triples. The diffusion is irreducible and converges strongly in L^2 to equilibrium. For 0<κ<4, the complement of the conformally removable welding locus has zero capacity.
ER  -