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Markov processes: Birth, reversal, regeneration. (Processus de Markov: Naissance, retournement, régénération.) (French) Zbl 0844.60042
Hennequin, P. L. (ed.), Ecole d’Eté de probabilités de Saint-Flour XXI - 1991, du 18 Août au 4 Septembre, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1541, 261-292 (1993).
These are lecture notes based on a course given by the author at the Saint Flour 1991 Summer School. The material presented is covered by the fifth volume of “Probabilités et potentiel”, in whose preparation the author has joined C. Dellacherie and P.-A. Meyer. The main theme is Markov processes with random birth and death whose transition semigroup is a given right semigroup and whose one-dimensional distribution is given by either an entrance law or by an excessive measure. Those processes were first introduced by Dynkin and Kuznetsov, and the results were adapted to the “right” setting by P. J. Fitzsimmons and the author [Probab. Theory Relat. Fields 72, No. 3, 319-336 (1986; Zbl 0547.60077)]. Since then they have attracted the attention of many authors in the field of Markov processes and potential theory. This course provides a very beautiful presentation of some of those results.
Among other things, the author treats the construction of processes with random birth and death (Kuznetsov processes). He studies the excessive measures that govern their one-dimensional distribution and obtains various decompositions such as their Riesz decomposition and their invariant and purely excessive parts, by splitting the path space \(W\) according to what happens at the birth time \((\{\alpha = -\infty\}\) for the invariant part, \(\{X_\alpha \in E\}\) for the potential in the Riesz decomposition). He treats time reversal of these processes and applies the results to processes in strong duality and the study of their excessive functions. He considers homogeneous random measures and their Revuz measures. There are strong connections between time reversal in the context of Kuznetsov processes and Azéma’s work on time reversal, which are discussed in the time reversal section. The last section is devoted to regeneration and exit systems. Excursion laws provide examples of laws of Markov processes with one-dimensional distribution that are entrance laws. The special case of excursions from a point is applied to a solution of a Skorokhod representation problem that was treated by Bertoin and Le Jan.
In my opinion, these lecture notes make one of the nicest presentations of the theory of processes with random birth and death.
For the entire collection see [Zbl 0778.00027].

60J25 Continuous-time Markov processes on general state spaces
60J40 Right processes
60J45 Probabilistic potential theory