2018/11/30 by Sergey Bezuglyi, Bezuglyi, Sergey, Palle E. T. Jørgensen +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1812.00081
openalex publication_date 2018/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (V, \mathcal B), such that for every x ∈ V , P(x; ⋅) is a probability measure on (V, \mathcal B). 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 \mathcal HE, 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 \mathcal HE, called finite-energy harmonic functions. A second reason for \mathcal HE is that it plays a key role in our introduction of a generalized Greens function. (The latter stands in relation to our present measure theoretic Laplace operator in a way that parallels more traditional settings of Greens functions from classical potential theory.) A third reason for \mathcal HE is its use in our analysis of path-space dynamics for generalized Markov transition systems.