Book chapter
Symmetric Measures, Continuous Networks, and Dynamics
New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative, pp.139-197
Operator Theory: Advances and Applications, Springer International Publishing
05/21/2021
DOI: 10.1007/978-3-030-76473-9_6
Abstract
With view to applications, we here give an explicit correspondence between the following two: (i) the set of symmetric and positive measures ρ on one hand, and (ii) a certain family of generalized Markov transition measures P, with their associated Markov random walk models, on the other. By a generalized Markov transition measure we mean a measurable and measure-valued function P on , such that for every x ∈ V, P(x;⋅) is a probability measure on ). Hence, with the use of our correspondence (i)–(ii), we study generalized Markov transitions P and path-space dynamics. Given P, we introduce an associated operator, also denoted by P, and we analyze its spectral theoretic properties with reference to a system of precise L2 spaces.
Our setting is more general than that of earlier treatments of reversible Markov processes. In a potential theoretic analysis of our processes, we introduce and study an associated energy Hilbert space , not directly linked to the initial L2-spaces. Its properties are subtle, and our applications include a study of the P-harmonic functions. They may be in , called finite-energy harmonic functions. A second reason for is that it plays a key role in our introduction of a generalized Green function. (The latter stands in relation to our present measure theoretic Laplace operator in a way that parallels more traditional settings of Green functions from classical potential theory.) A third reason for is its use in our analysis of path-space dynamics for generalized Markov transition systems.
Details
- Title: Subtitle
- Symmetric Measures, Continuous Networks, and Dynamics
- Creators
- Sergey Bezuglyi - University of IowaPalle E. T Jorgensen - University of Iowa
- Resource Type
- Book chapter
- Publication Details
- New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative, pp.139-197
- Publisher
- Springer International Publishing; Cham
- Series
- Operator Theory: Advances and Applications
- DOI
- 10.1007/978-3-030-76473-9_6
- eISSN
- 2296-4878
- ISSN
- 0255-0156
- Language
- English
- Date published
- 05/21/2021
- Academic Unit
- Mathematics
- Record Identifier
- 9984241041202771
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