## The Annals of Probability

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

### Symmetry Groups and Translation Invariant Representations of Markov Processes

#### 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