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Stochastic processes induced by singular operators
TL;DR: In this paper, a general family of multivariable Gaussian stochastic processes is studied, where each process is prescribed by a fixed Borel measure π on π R^n.
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Abstract: In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure $\sigma$ on $\mathbb R^n$. The case when $\sigma$ is assumed absolutely continuous with respect to Lebesgue measure was studied earlier in the literature, when $n=1$. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures $\sigma$ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in Hilbert space, and their spectral theory.
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Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis
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The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative
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TL;DR: In this paper, the authors give necessary and sufficient conditions for a continuous positive definite function to have exactly one extension and prove that each such unitary representation has a simple spectrum, which has applications to Gaussian stochastic processes.
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Graph laplacians and discrete reproducing kernel hilbert spaces from restrictions and cameron-martin
Palle E. T. Jorgensen,Feng Tian +1 more
TL;DR: In this article, reproducing kernel functions and associated Reproducing kernel Hilbert spaces (RKHSs) H over infinite, discrete and countable sets V were studied for finding approximate solutions to PDE-boundary value problems; typically using multiresolution-subdivision schemes, applied to the continuous domains.
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Self-adjoint extensions of network Laplacians and applications to resistance metrics
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Integration questions related to fractional Brownian motion
Vladas Pipiras,Murad S. Taqqu +1 more
TL;DR: In this paper, it was shown that the reproducing kernel Hilbert space of fractional Brownian motion integrands can be characterized by elementary functions, and a similar characterization can be obtained when H ∈ (0, 1/2) or H∈ (1/2, 1).
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