Two generalizations of Mehler's formula in white noise analysis. Issue 4 (19th May 2023)

• Record Type: Journal Article
• Authors: Bock, Wolfgang; Bock, Maximilian
• Is Part Of: Stochastics. Volume 95:Issue 4(2023)
• Page Start: 501
• Page End: 520
• Publication Date: 2023-05-19
• Subjects: White noise analysis -- Mehler's formula -- gross laplacian -- number operator -- Fourier–Gauss transform -- generalized Wick tensors; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2022.2089039
• Languages: English
• ISSNs: 1744-2508

Abstract

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.