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Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity — arXiv:2610.06341

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.06341
License (as archived): CC0-1.0
Archived: 2026-10-06T13:48:38.551Z
SHA-256 of the archived text: 1abf3e2f9b5d5622dcdd54b69f7f188ea5b43e1ebc13f8f83ba440af8bfb4f8a
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How to cite

Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity — arXiv:2610.06341 — arXiv (en), https://arxiv.org/abs/2610.06341 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fcffd4acc90a (Metraipolis archive dc_fcffd4acc90a, sha256 1abf3e2f9b5d5622dcdd54b69f7f188ea5b43e1ebc13f8f83ba440af8bfb4f8a)

BibTeX

@misc{dcfcffd4acc90a,
  title = {Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity — arXiv:2610.06341},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.06341},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fcffd4acc90a; sha256 1abf3e2f9b5d5622dcdd54b69f7f188ea5b43e1ebc13f8f83ba440af8bfb4f8a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.06341. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension $d\geq2$. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free $L^\infty_t{\rm BMO}^{-1}_x$ drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in $L^\infty_tL^d_x$. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent $L^{d,\infty}$ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent $L^d$ drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2610.06341 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T12:06:32.174Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).