The authors present a review of the main properties of reciprocal processes. Their measure-reciprocal approach allows for a unified treatment of diffusion and jump processes.
The paper is divided into three sections: Time-symmetry of Markov measures, Reciprocal measures and Reciprocal measures are solutions of entropy minimizing problems.
Section 1 is devoted to the structure of Markov measures and to their time-symmetries. Then, in Section 2, reciprocal measures are introduced and their relationship with Markov measures is investigated. Finally, in Section 3, the authors sketch the tight connection between reciprocal classes and sonic specific entropy minimization problems.
An extensive list of references is given at the end of the paper.
After a short historical presentation in which the reader is reminded that the Markov property is a standard probabilistic notion since its formalization at the beginning of the 20th century, the authors refer to a paper of B. Jamison [Z. Wahrscheinlichkeitstheor. Verw. Geb. 30, 65–86 (1974; Zbl 0326.60033)] and provide some new results. They present a unifying measure-theoretical approach to reciprocal processes. Unlike Jamison, who worked with finite-dimensional distributions using the concept of reciprocal transition probability function, they look at Markov and reciprocal processes as path measures, i.e., measures on the path space.
The authors also provide a possible extension. Thus, it is emphasized that they focus on probability path measures. As a consequence, they drop the word probability: any path probability measure, Markov probability measure or reciprocal probability measure is simply called a path measure, a Markov measure or a reciprocal measure.
The results can easily be extended to \(\sigma\)-finite path measures, e.g., processes admitting an unbounded measure as their initial law.