Most of the papers compiled in this volume have been published in Uspekhi Matematicheskikh Nauk and translated into English in the Russian Mathematical Surveys. The core consists of the series [IV], [V], [VI], [VII] presenting a new approach to Markov processes (especially to the Martin boundary theory and the theory of duality) with the following distinctive features:

1. The general non-homogeneous theory precedes the homogeneous one. This is natural because non-homogeneous Markov processes are invariant with respect to all monotone transformations of time scale – a property which is destroyed in the homogeneous case by the introduction of an additional structure: a one-parameter semi-group of shifts. In homogeneous theory, the probabilistic picture is often obscured by the technique of Laplace transforms.

2. All the theory is invariant with respect to time reversion. We consider processes with random birth and death times and we use on equal terms the forward and backward transition probabilities, i.e., the conditional probability distributions of the future after t and of the past before t given the state at time t. (This is an alternative to introducing a pair of processes in duality defined on different sample spaces.)

3. […]