In 1972, starting from a transient Markov process with a nice semigroup satisfying the absolute continuity hypothesis, P. A. Meyer and I built a nice dual semigroup and then obtained a Martin compactification modulo a polar set. Now, in this paper, we start from this Martin space and study the behavior of the sample paths. We prove that the Martin boundary so constructed appears in the classical form which allows one to describe the final behavior of the sample paths. We also prove that the Martin boundary we construct is an "entrance boundary" such as the Ray boundary. Finally we study a class of additive functionals which ignore the discontinuities of the process in the Martin space and which constitute a nice class of "natural" additive functionals. From all this we conclude that our Martin space is better suited for the study of the process than either the Ray space or the original space.
"Une Theorie de la Dualite a Ensemble Polaire Pres II." Ann. Probab. 4 (6) 947 - 976, December, 1976. https://doi.org/10.1214/aop/1176995939