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Symmetric Measures, Continuous Networks, and Dynamics

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New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative

Part of the book series: Operator Theory: Advances and Applications ((LOLS,volume 286))

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 L 2 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 L 2-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.

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Acknowledgements

The authors are thankful to colleagues and collaborators, especially the members of the seminars in Mathematical Physics and Operator Theory at the University of Iowa, where versions of this work have been presented. We acknowledge very helpful conversations with among others Professors Paul Muhly, Wayne Polyzou; and conversations at distance with Professors Daniel Alpay, and his colleagues at both Ben Gurion University, and Chapman University. We wish to thank the referee for useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa, IA, USA

    Sergey Bezuglyi & Palle E. T. Jorgensen

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  1. Sergey Bezuglyi
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  2. Palle E. T. Jorgensen
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Correspondence to Palle E. T. Jorgensen .

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Editors and Affiliations

  1. Schmid College of Science & Technology, Chapman University, Orange, CA, USA

    Daniel Alpay

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Ronen Peretz

  3. Faculty of Sciences, Holon Institute of Technology, Holon, Israel

    David Shoikhet

  4. Schmid College of Science and Technology, Chapman University, Orange, CA, USA

    Mihaela B. Vajiac

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Bezuglyi, S., Jorgensen, P.E.T. (2021). Symmetric Measures, Continuous Networks, and Dynamics. In: Alpay, D., Peretz, R., Shoikhet, D., Vajiac, M.B. (eds) New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative. Operator Theory: Advances and Applications(), vol 286. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-76473-9_6

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