The Annals of Probability

Symmetry Groups and Translation Invariant Representations of Markov Processes

Joseph Glover

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Abstract

The symmetry groups of the potential theory of a Markov process $X_t$ are used to introduce new algebraic and topological structures on the state space and the process. For example, let $G$ be the collection of bijections $\varphi$ on $E$ which preserve the collection of excessive functions. Assume there is a transitive subgroup $H$ of the symmetry group $G$ such that the only map $\varphi \in H$ fixing a point $e \in E$ is the identity map on $E$. There is a bijection $\Psi: E \rightarrow H$ so that the algebraic structure of $H$ can be carried to $E$, making $E$ into a group. If there is a left quasi-invariant measure on $E$, then there is a topology on $E$ making $E$ into a locally compact second countable metric group. There is also a time change $\tau(t)$ of $X_t$ such that $X_{\tau(t)}$ is a translation invariant process on $E$ and $X_{\tau(t)}$ is right-continuous with left limits in the new topology.

Article information

Source
Ann. Probab., Volume 19, Number 2 (1991), 562-586.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990441

Digital Object Identifier
doi:10.1214/aop/1176990441

Mathematical Reviews number (MathSciNet)
MR1106276

Zentralblatt MATH identifier
0732.60079

JSTOR
links.jstor.org

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Markov process potential theory topological groups Lie groups

Citation

Glover, Joseph. Symmetry Groups and Translation Invariant Representations of Markov Processes. Ann. Probab. 19 (1991), no. 2, 562--586. doi:10.1214/aop/1176990441. https://projecteuclid.org/euclid.aop/1176990441


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