On the one hand, in almost all the scientific areas, from physical to social sciences, biology to economics, from meteorology to pattern recognition in remote sensing, the theory of classical probability plays a major role and on the other much of our knowledge about the physical world at least is based on the quantum theory [12]. In a way, quantum theory itself is a new kind of theory of probability (in the language of von Neumann and Birkhoff) (see for example [106]) which contains the classical model, and therefore it is natural to extend the other areas of classical probability theory, in particular the theory of Markov processes and stochastic calculus to this quantum model.

There are more than one possible ways (see for example [127]) to construct the above-mentioned extension and in this book we have chosen the one closest to the classical model in spirit, namely that which contains the classical theory as a submodel. This requirement has ruled out any discussion of areas such as free and monotone-probability models. Once we accept this quantum probabilistic model, the ‘grand design’ that engages us is the ‘canonical construction of a *-homomorphic flow (satisfying a suitable differential equation) on a given algebra of observables such that the expectation semigroup is precisely the given contractive semigroup of completely positive maps on the said algebra’.

This problem of ‘dilation’ is here solved completely for the case when the semigroup has a bounded generator, and also for the more general case (of an unbounded generator) with certain additional conditions such as symmetry and/or covariance with respect to a Lie group action.