波长尺度的可选停止、临界Feynman-Kac规范与容量谱不等式
Wavelength-scale optional stopping, critical Feynman-Kac gauges, and capacitary spectral inequalities
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中文总结 AI 辅助
该研究发展两种布朗运动停止方法，结合尺度得到热观测性与零可控性，推导Feynman-Kac鞅平方的性质、本征函数相关不等式及估计，为相关谱与控制问题提供理论结果。
中文摘要 AI 辅助
我们发展了两种布朗运动停止方法：一种利用击中概率与容量，另一种利用加权退出分布的矩。在闭的m维黎曼流形（m≥2）上，第一种方法给出了支撑在零体积集合上的平衡测度的低能谱不等式。结合尺度可得到来自Hausdorff维数为m-2的稠密集所承载的全支撑测度的热观测性与零可控性，该测度在有界H¹迹条件下是极小的。对于每个非递减无界的Φ且满足Φ(4E)≤DΦ(E)，可选择该测度使得最优谱常数与Φ(E)相当，短时控制成本为T^{-1/2}Φ(T^{-1})^{1/2}。将Feynman-Kac鞅平方会使势加倍：对于径向V，只要(½Δ+2V)g=0的正则解保持正，边界二阶矩在log r中是对数凸的，其中r²V(r)非递减；当V(0)>0时，因子2是最优的。在小测地球上，加权二阶矩给出几乎单调的频率和加权三半径不等式；其对数导数的界给出加倍估计，经基态变换后可用于Logunov-Malinnikova的Remez定理。归一化一阶矩退出律给出拉普拉斯本征函数的平均边界方差恒等式。在r=cλ^{-1/2}处，在承载φ²质量的O(c^{2/m})的集合之外，{φ(x)φ>0}∩B(x,r)中过x的分量包含一个同心的近全半径球，几乎填满B(x,r)，且其第一狄利克雷本征值几乎匹配。一阶矩停止也控制正的超级水平分量。反射布朗运动与局部时给出有界Lipschitz区域上Steklov本征函数的边界到领圈L^∞估计，以及C²区域上的L^2估计。
英文摘要
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 $Φ$ with $Φ(4E)\leq DΦ(E)$, the measure can be chosen so that the optimal spectral constant is comparable to $Φ(E)$ and the small-time control cost to $T^{-1/2}Φ(T^{-1})^{1/2}$. 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 $(\tfrac12Δ+2V)g=0$ 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 $r=cλ^{-1/2}$, outside a set carrying $O(c^{2/m})$ of the $φ^2$-mass, the component of $\{φ(x)φ>0\}\cap B(x,r)$ 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.
