横截相交的Kac-Rice公式
Kac-Rice formula for transverse intersections
摘要：
我们证明了一个广义的卡茨-赖斯公式，在一个定义良好的正规设置，计算期望的基数的一个子流形的原像通过随机映射，通过将其表示为密度的积分。我们的证明是从零开始的，虽然遵循了Kac-Rice公式的标准证明的指导思想，但从测度论的角度出发，包含了一些新的思想.进一步推广，我们将这个公式扩展到任何其他类型的计数测度，如相交度。深入讨论了向量丛的高斯随机截面的光滑化问题。这里，该公式计算截面满足全空间的给定子流形的期望点数，它在自然非退化条件下成立，并且可以通过使用适当的连接来简化。此外，我们指出了一类子流形，我们称之为次高斯，其中的公式是局部有限的，并连续依赖于第一个喷流的协方差。特别是，这适用于任何概念的奇点的部分，可以定义为一组点，其中喷流延长满足一个给定的半代数子流形的喷流空间。各种应用和特殊情况下的例子进行了讨论。特别是，我们报告了一个新的证明的庞加莱运动公式的齐性空间，我们观察如何简化公式的各向同性高斯领域的球。
英文摘要：
We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map, by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere.
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