The Annals of Probability

Symmetry Groups and Translation Invariant Representations of Markov Processes

Joseph Glover

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Markov process potential theory topological groups Lie groups


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

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