SYMMETRY GROUPS AND TRANSLATION INVARIANT REPRESENTATIONS OF MARKOV-PROCESSES

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 phi 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 phi-epsilon-H fixing a point e epsilon-E is the identity map on E. There is a bijection PSI: E --> 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.