Description
The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In the Hilbert case, their decomposition is useful for simulation and data analysis (Principal Component Analysis, PCA). This relies on a spectral decomposition of the covariance operator, which provides a canonical decomposition of Gaussian measures on Hilbert spaces, the so-called Karhunen-Loève expansion. In this work, we intend to extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated to Gaussian measures on Banach spaces. In the special case of the standard Wiener measure, this decomposition matches with Paul Lévy’s construction of Brownian motion.
