Geodesic
Construction of a Markov, space homogeneous, non-death process from hitting distributions
Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 207-224
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A sufficient condition for the existence of a space homogeneous, non-death strong Markov process with given hitting probabilities is obtained. 
@article{ZNSL_1979_85_a17,
     author = {B. P. Harlamov and V. E. Janimiagi},
     title = {Construction of {a~Markov,} space homogeneous, non-death process from hitting distributions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {207--224},
     year = {1979},
     volume = {85},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a17/}
}
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B. P. Harlamov; V. E. Janimiagi. Construction of a Markov, space homogeneous, non-death process from hitting distributions. Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 207-224. http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a17/