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Metric Duality Between Positive Definite Kernels and Boundary Processes
Journal article   Peer reviewed

Metric Duality Between Positive Definite Kernels and Boundary Processes

Palle Jorgensen and Feng Tian
International Journal of Applied and Computational Mathematics, Vol.4(1), pp.1-13
02/2018
DOI: 10.1007/s40819-017-0434-1

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Abstract

We study representations of positive definite kernels K in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general given positive definite kernel, but are then subsequently specialized to the above mentioned applications. Given a positive definite kernel K on $$S\times S$$ S×S where S is a fixed set, we first study families of factorizations of K. By a factorization (or representation) we mean a probability space $$\left( B,\mu \right) $$ B,μ and an associated stochastic process indexed by S which has K as its covariance kernel. For each realization we identify a co-isometric transform from $$L^{2}\left( \mu \right) $$ L2μ onto $$\mathscr {H}\left( K\right) $$ HK , where $$\mathscr {H}\left( K\right) $$ HK denotes the reproducing kernel Hilbert space of K. In some cases, this entails a certain renormalization of K. Our emphasis is on such realizations which are minimal in a sense we make precise. By minimal we mean roughly that B may be realized as a certain K-boundary of the given set S. We prove existence of minimal realizations in a general setting.
 Harmonic Analysis Mathematics 65R10 Reproducing kernel Hilbert space Path-space measures Theoretical, Mathematical and Computational Physics Transforms Computational Science and Engineering 46N50 Covariance Operations Research/Decision Theory Nuclear Energy Hilbert space Applications of Mathematics Mathematical Modeling and Industrial Mathematics Gaussian free fields Primary 47L60 42C15 46N30

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