  We deal with the general structure of (noncommutative) stochastic processes
by using the standard techniques of Operator Algebras. Any stochastic process
is associated to a state on a universal object, i.e. the free product
$C^*$-algebra in a natural way. In this setting one recovers the classical
(i.e. commutative) probability scheme and many others, like those associated to
the Monotone, Boolean and the $q$-deformed canonical commutation relations
including the Bose/Fermi and Boltzmann cases. Natural symmetries like
stationarity and exchangeability, as well as the ergodic properties of the
stochastic processes are reviewed in detail for many interesting cases arising
from Quantum Physics and Probability.
