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

2022/05/16 by Jorgensen, Palle E. T., Tian, James
#46N20. Secondary: 46E22 #46N30 #46N50 #47B25 #47E05 #60G15 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47B32

paper · doi:10.48550/arxiv.2205.07645

Abstract

We show that a Krein-Feller operator is naturally associated to a fixed measure μ, assumed positive, σ-finite, and non-atomic. Dual pairs of operators are introduced, carried by the two Hilbert spaces, L²(μ) and L²(λ), where λ 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 μ-Krein-Feller operator as a closable quadratic form. As an application, for a given measure μ, including the case of fractal measures, we compute the associated diffusion, semigroup, Dirichlet forms, and μ-generalized heat equation.

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