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.