Around Tsirelson's equation, or: The evolution process may not explain everything

We present a synthesis of a number of developments which have been made around the celebrated Tsirelson's equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $ G $, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $ G=\bT $.