闭式响应演算与Liouville量子引力度量的联合密度
Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric
- School of Mathematics (Zhuhai), Sun Yat-Sen University（中山大学数学学院（珠海））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种将底层概率空间的一阶响应转化为可观测分布上闭式微分演算的一般准则，并应用于次临界Liouville量子引力度量，证明对数距离比向量具有Lebesgue密度。
AI中文摘要：
我们给出了一个一般准则，用于将底层概率空间上的一阶响应转移到可观测量的分布上的闭式微分演算。一个分部积分恒等式消除了表示歧义，并产生了一个可闭的梯度、其散度以及一个闭的Markov形式。我们将此方案应用于整个$0<γ<2$范围内的次临界Liouville量子引力度量。Weyl扰动测地线的序列紧性将对数距离比的导数与两个归一化测地线占据测度之差的Sobolev Riesz代表元等同起来。投影度量律上的诱导形式是高斯方向形式的像，并具有能量-像-密度性质。最后，固定目标合流和叶消除论证使得每个有限标记对森林的响应Gram矩阵正定。因此，相应的对数距离比向量具有Lebesgue密度。
英文摘要：
We give a general criterion for transferring first-order responses on an underlying probability space to a closed differential calculus on the law of an observable. An integration-by-parts identity removes presentation ambiguity and yields a closable gradient, its divergence, and a closed Markov form. We apply this scheme to the subcritical Liouville quantum gravity metric throughout the range $0<γ<2$. Sequential compactness of Weyl-perturbed geodesics identifies the derivative of a logarithmic distance ratio with the Sobolev Riesz representative of the difference of two normalized geodesic occupation measures. The induced form on the projective metric law is the image of a Gaussian directional form and has the energy-image-density property. Finally, fixed-target confluence and a leaf-elimination argument make the response Gram matrix positive definite for every finite forest of marked pairs. The corresponding vector of logarithmic distance ratios therefore has a Lebesgue density.
