Geodesic
Continuous One-Dimensional Markov Processes and Additive Functionals Derived from Them
Teoriâ veroâtnostej i ee primeneniâ, Tome 4 (1959) no. 2, pp. 208-211

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The terminology and symbols employed are the same as in [1] and [2]. The paper describes all continuous Feller one-dimensional Markov processes, which are regular within a segment and gives their infinitesimal operators (Theorem 4). It is stated that each such process is a sub-process of a non-terminating process of the same type (Theorems and 1'). Therefore, a description of all processes amounts to a description of all multiplicative or additive functionals corresponding to regular Feller sub-processes of non-terminating processes. 
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     author = {V. A. Volkonskii},
     title = {Continuous {One-Dimensional} {Markov} {Processes} and {Additive} {Functionals} {Derived} from {Them}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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V. A. Volkonskii. Continuous One-Dimensional Markov Processes and Additive Functionals Derived from Them. Teoriâ veroâtnostej i ee primeneniâ, Tome 4 (1959) no. 2, pp. 208-211. http://geodesic.mathdoc.fr/item/TVP_1959_4_2_a5/