ABSTRACT

Mehler's formula is an important tool in Gaussian analysis. In this article, we study two generalizations of Mehler's formula for the Ornstein–Uhlenbeck semigroup, i.e. the semigroup generated by the number operator. The first generalization leads to transformation groups which have as infinitesimal generator a perturbation of the number operator with suitable integral kernel operators, which are well studied in white noise analysis. For the second one, we characterize the complex Hida measures for which a version of Mehler's formula for the Ornstein–Uhlenbeck semigroup can be extended to. We apply this result to the Feynman integrand for a quadratic potential. Here the time independent eigenstates of the considered transformation groups and the time evolution of eigenvalues are provided.

摘要

梅勒公式是高斯分析中的重要工具。在本文中，我们研究了 Mehler 公式对 Ornstein–Uhlenbeck 半群的两个推广，即由数算子生成的半群。第一个推广导致变换群具有作为无穷小生成器的数算子的扰动以及合适的积分核算子，这在白噪声分析中得到了很好的研究。对于第二个，我们描述了复杂的 Hida 测度，可以将 Mehler 公式的一个版本扩展到 Ornstein-Uhlenbeck 半群。我们将此结果应用于费曼被积函数以获得二次势。这里提供了所考虑的变换群的时间独立本征态和本征值的时间演化。