Wavelength-scale optional stopping, critical Feynman-Kac gauges, and capacitary spectral inequalities

Abstract

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, $m\ge 2$, 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 $\Phi (4E)\le D\Phi (E)$, 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 $r^{2}V(r)$ is nondecreasing, wherever the regular solution of $(\frac{1}{2}\Delta +2V)g=0$ 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 ${\phi }^2$-mass, the component of $\{\phi (x)\phi >0\}\cap B(x,r)$ 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.

67 pages, comments welcome!