Geodesic
Additive functionals of semi-Markov processes with absorption
Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 329-339.

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Given a semi-Markov process with an arbitrary set of states, a criterion is obtained for the attainability of a certain isolated subset of states and for finiteness of the average attainment time. An equation is given for the average of an additive functional of a process with absorption, existence and uniqueness conditions are deduced for the solution of that equation in a given class of functions, and an integral representation is obtained for the solution. 
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     author = {A. P. Cherenkov},
     title = {Additive functionals of {semi-Markov} processes with absorption},
     journal = {Matemati\v{c}eskie zametki},
     pages = {329--339},
     publisher = {mathdoc},
     volume = {17},
     number = {2},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a15/}
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A. P. Cherenkov. Additive functionals of semi-Markov processes with absorption. Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 329-339. http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a15/