Journal article
Realizations and Factorizations of Positive Definite Kernels
Journal of Theoretical Probability, Vol.32(4), pp.1925-1942
12/2019
DOI: 10.1007/s10959-018-0868-3
Abstract
Given a fixed sigma-finite measure space $$\left( X,\mathscr {B},\nu \right) $$ X , B , ν , 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 $$\nu $$ ν is infinite, but sigma-finite. We introduce such positive definite kernels $$K\left( \cdot ,\cdot \right) $$ K · , · with the two variables from the subclass of the sigma-algebra $$\mathscr {B}$$ B whose elements are sets with finite $$\nu $$ ν 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 $$L^{2}\left( \nu \right) $$ L 2 ν . 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.
Details
- Title: Subtitle
- Realizations and Factorizations of Positive Definite Kernels
- Creators
- Palle Jorgensen - grid.214572.7 0000 0004 1936 8294 The University of Iowa Iowa City IA 52242-1419 USAFeng Tian - grid.256774.5 0000 0001 2322 3563 Hampton University Hampton VA 23668 USA
- Resource Type
- Journal article
- Publication Details
- Journal of Theoretical Probability, Vol.32(4), pp.1925-1942
- DOI
- 10.1007/s10959-018-0868-3
- ISSN
- 0894-9840
- eISSN
- 1572-9230
- Publisher
- Springer US; New York
- Language
- English
- Date published
- 12/2019
- Academic Unit
- Mathematics
- Record Identifier
- 9983985995702771
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