Realizations and Factorizations of Positive Definite Kernels

Abstract

Given a fixed sigma-finite measure space \(\left( X,\mathscr {B},\nu \right) \), we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure \(\nu \) is infinite, but sigma-finite. We introduce such positive definite kernels \(K\left( \cdot ,\cdot \right) \) with the two variables from the subclass of the sigma-algebra \(\mathscr {B}\) whose elements are sets with finite \(\nu \) measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in \(L^{2}\left( \nu \right) \). Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.

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Acknowledgements

The co-authors thank colleagues for helpful and enlightening discussions, and members in the Math Physics seminar at The University of Iowa.

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Correspondence to Feng Tian.

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Jorgensen, P., Tian, F. Realizations and Factorizations of Positive Definite Kernels. J Theor Probab 32, 1925–1942 (2019). https://doi.org/10.1007/s10959-018-0868-3

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Keywords

  • Reproducing kernel Hilbert space
  • Harmonic analysis
  • Gaussian free fields
  • Covariance
  • Generalized Ito integration

Mathematics Subject Classification (2010)

  • Primary 47L60
  • 46N30
  • 65R10
  • 42C15
  • 31C20