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Wavelength-scale optional stopping, critical Feynman-Kac gauges, and capacitary spectral inequalities — arXiv:2608.15650

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.15650
License (as archived): CC0-1.0
Archived: 2026-10-18T20:25:37.704Z
SHA-256 of the archived text: 4eb7fa055f664dfe3398341f2b0099eba125c53cc89f13e8149e9f75f8188461
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How to cite

Wavelength-scale optional stopping, critical Feynman-Kac gauges, and capacitary spectral inequalities — arXiv:2608.15650 — arXiv (en), https://arxiv.org/abs/2608.15650 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e2ecd6735014 (Metraipolis archive dc_e2ecd6735014, sha256 4eb7fa055f664dfe3398341f2b0099eba125c53cc89f13e8149e9f75f8188461)

BibTeX

@misc{dce2ecd6735014,
  title = {Wavelength-scale optional stopping, critical Feynman-Kac gauges, and capacitary spectral inequalities — arXiv:2608.15650},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.15650},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e2ecd6735014; sha256 4eb7fa055f664dfe3398341f2b0099eba125c53cc89f13e8149e9f75f8188461},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.15650. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We develop two Brownian stopping methods: one uses hitting probabilities and capacity, the other moments of weighted exit distributions. On a closed $m$-dimensional Riemannian manifold, $m\geq 2$, the first gives low-energy spectral inequalities for equilibrium measures supported on sets of zero volume. Combining scales gives heat observability and null controllability from a full-support measure carried by a dense set of Hausdorff dimension $m-2$, minimal under a bounded $H^1$-trace condition. For every nondecreasing unbounded $\Phi$ with $\Phi(4E)\leq D\Phi(E)$, the measure can be chosen so that the optimal spectral constant is comparable to $\Phi(E)$ and the small-time control cost to $T^{-1/2}\Phi(T^{-1})^{1/2}$. Squaring the Feynman-Kac martingale doubles the potential: for radial $V$, the boundary second moment is log-convex in $\log r$ if $r^2V(r)$ is nondecreasing, wherever the regular solution of $(\tfrac12\Delta+2V)g=0$ stays positive; the factor $2$ is sharp when $V(0)>0$. On small geodesic balls, weighted second moments give an almost-monotone frequency and a weighted three-radius inequality; bounds for its logarithmic derivative give doubling estimates, which, after a ground-state transform, feed the Remez theorem of Logunov-Malinnikova. The normalised first-moment exit law gives an averaged boundary-variance identity for Laplace eigenfunctions. At $r=c\lambda^{-1/2}$, outside a set carrying $O(c^{2/m})$ of the $\varphi^2$-mass, the component of $\{\varphi(x)\varphi>0\}\cap B(x,r)$ through $x$ contains a concentric ball of nearly full radius, fills nearly all of $B(x,r)$, and nearly matches its first Dirichlet eigenvalue. First-moment stopping also controls positive superlevel components. Reflected Brownian motion and local time give boundary-to-collar $L^\infty$ estimates for Steklov eigenfunctions on bounded Lipschitz domains, and an $L^2$ estimate for $C^2$ domains.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-18 in math.PR).
Categorie arXiv: Analysis of PDEs (math.AP) ; Probability (math.PR); Spectral Theory (math.SP).
Pagina del lavoro: https://arxiv.org/abs/2608.15650 — 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 2026-10-18T19:45:36.395Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-18?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).