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Karhunen-Loève Decomposition of Gaussian Measures on Banach Spaces

Bay Xavier, Croix Jean-Charles

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 2

279--297

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectra theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen-Loève expansion. In this paper, we 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 with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy-Ciesielski construction of Brownian motion.