Abstract
Algebraic structures have enjoyed a special role in Markov processes from the very beginning of the subject. Probabilists focussed much of their attention initially on independent increment processes in the group R. Since these processes have convolution semigroups, they are especially amenable to concrete analysis by Fourier techniques. As Markov processes matured, probabilists began to study them in more general settings, and there is often no algebraic structure in evidence on the state space these days. In this article, we will explore methods of introducing algebraic structures on the state space which are naturally associated with the process. In several instances, after an invertible probabilistic transformation, they will become independent increment processes in the new group structure.
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