Book chapter
Stochastics and Dynamics of Fractals
Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis, pp.171-216
Operator Theory: Advances and Applications, 290, Springer International Publishing
2023
DOI: 10.1007/978-3-031-21460-8_5
Abstract
For classes of self-affine fractal measures, we offer a new analysis of the associated diffusions, Markov processes, semigroups, and Dirichlet forms. With view to an harmonic analysis of fractal measures, we introduce particular dual-pair systems of operators. The starting point is a fixed measure μ. We then introduce associated transforms which allow a comparison of μ to Lebesgue measure λ. Our dual pairs are carried by the two Hilbert spaces, L2(μ ) and L2(λ). A dual-pair analysis consist of a choice of a pair of Hilbert spaces, and two densely defined (unbounded) operators, each one contained in the adjoint of the other. Applications include the Krein-Feller operator, and the associated μ diffusion. Our approach goes beyond earlier related results in the general theory, in several ways: For example, our new duality framework is versatile, and it accommodates the setting of general measure spaces, thus going beyond that of classical domains, or IFSs, in ambient ℝk. Key tools for our analysis are associated reproducing kernel Hilbert spaces (RKHSs), time-change, and Gaussian fields.
Details
- Title: Subtitle
- Stochastics and Dynamics of Fractals
- Creators
- Palle E. T. JorgensenJames Tian
- Resource Type
- Book chapter
- Publication Details
- Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis, pp.171-216
- Publisher
- Springer International Publishing; Cham
- Series
- Operator Theory: Advances and Applications; 290
- DOI
- 10.1007/978-3-031-21460-8_5
- eISSN
- 2296-4878
- ISSN
- 0255-0156
- Language
- English
- Date published
- 2023
- Academic Unit
- Mathematics
- Record Identifier
- 9984388755102771
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