Book contents
- Frontmatter
- Contents
- The asymptotic speed and shape of a particle system
- On doubly stochastic population processes
- On limit theorems for occupation times
- The Martin boundary of two dimensional Ornstein-Uhlenbeck processes
- Green's and Dirichlet spaces for a symmetric Markov transition function
- On a theorem of Kabanov, Liptser and Širjaev
- Oxford Commemoration Ball
- Invariant measures and the q-matrix
- The appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes
- Three unsolved problems in discrete Markov theory
- The electrostatic capacity of an ellipsoid
- Stationary one-dimensional Markov random fields with a continuous state space
- A uniform central limit theorem for partial-sum processes indexed by sets
- Multidimensional randomness
- Some properties of a test for multimodality based on kernel density estimates
- Criteria for rates of convergence of Markov chains, with application to queueing and storage theory
- Competition and bottle-necks
- Contributors
Green's and Dirichlet spaces for a symmetric Markov transition function
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- The asymptotic speed and shape of a particle system
- On doubly stochastic population processes
- On limit theorems for occupation times
- The Martin boundary of two dimensional Ornstein-Uhlenbeck processes
- Green's and Dirichlet spaces for a symmetric Markov transition function
- On a theorem of Kabanov, Liptser and Širjaev
- Oxford Commemoration Ball
- Invariant measures and the q-matrix
- The appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes
- Three unsolved problems in discrete Markov theory
- The electrostatic capacity of an ellipsoid
- Stationary one-dimensional Markov random fields with a continuous state space
- A uniform central limit theorem for partial-sum processes indexed by sets
- Multidimensional randomness
- Some properties of a test for multimodality based on kernel density estimates
- Criteria for rates of convergence of Markov chains, with application to queueing and storage theory
- Competition and bottle-necks
- Contributors
Summary
Introduction
This is the first paper in a series devoted to Green's and Dirichlet spaces. In the next publications we shall study the spaces associated with fine Markov processes and with a certain class of multiparameter processes.
For the Brownian motion with exponential killing, the Dirichlet space is Sobolev's space H1 and Green's space is the dual space H−1. Both spaces are widely used in the theory of the free field (arising in quantum field theory). General Dirichlet and Green's spaces can be applied in an analogous way to Gaussian random fields associated with Markov processes [2].
Axiomatic theory of Dirichlet spaces was developed by Beurling and Deny [1]. Silverstein [5] and Fukushima [3] investigated the relation between Dirichlet spaces and Markov processes.
We start from a symmetric Markov transition function and we deal simultaneously with a pair: the Dirichlet space H and Green's space K. They are in a natural duality and they play symmetric roles but, in some respects, K is simpler than H. We consider several models for K and H. In particular, we represent them by L-valued functions of time t where L is a functional Hilbert space. We get the conventional representation of H by passage to the limit as t → ∞. Analogously, letting t → 0, we arrive at a representation of K by distributions (generalized functions).
- Type
- Chapter
- Information
- Probability, Statistics and Analysis , pp. 79 - 98Publisher: Cambridge University PressPrint publication year: 1983
- 5
- Cited by