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Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 205-248.

Dual pairs of operators, harmonic analysis of singular nonatomic measures and Krein-Feller diffusion

Authors:  Palle E.T. Jorgensen (1), James Tian (2)
Author institution: (1) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A.
(2) Mathematical Reviews, 416 4th Str. Ann Arbor, MI 48103-4816, U.S.A.

Summary: We show that a Krein--Feller operator is naturally associated to a fixed measure $\mu$, (positive, $\sigma$-finite, and nonatomic). Dual pairs of operators are introduced, carried by the two Hilbert spaces, $L^{2} (\mu )$ and $L^{2} (\lambda )$, where $\lambda$ denotes Lebesgue measure. An associated operator pair consists of two densely defined operators, each one contained in the adjoint of the other. This yields a rigorous analysis of the corresponding $\mu$-Krein--Feller operator. As an application, including the case of fractal measures, we compute the associated diffusion, semigroup, Dirichlet forms, and $\mu$-generalized heat equation.

Keywords: reproducing kernel Hilbert space, Gaussian free fields, generalized Ito integration, Krein--Feller operators, dual pairs, iterated function systems, selfadjoint extensions, Dirichlet forms

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