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arXiv · 2205.07645

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

Palle E. T. Jorgensen·James Tian
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Abstract

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

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Palle E. T. Jorgensen, James Tian. 2022-05-16. Dual pairs of operators, harmonic analysis of singular non-atomic measures and Krein-Feller diffusion. https://arxiv.org/abs/2205.07645

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