Geodesic
A~necessary and sufficient condition for the global-in-time existence of solutions to stochastic differential and parabolic equations on manifolds
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 69-76.

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A necessary and sufficient condition of one-sided type for the completeness of a stochastic flow and the corresponding diffusion semigroup on a manifold $M$ is found under the assumption that the space $C_0(M)$ of functions is invariant. 
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Yu. E. Gliklikh. A~necessary and sufficient condition for the global-in-time existence of solutions to stochastic differential and parabolic equations on manifolds. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 69-76. http://geodesic.mathdoc.fr/item/FPM_2007_13_8_a4/

[1] Gikhman I. I., Skorokhod A. V., Vvedenie v teoriyu stokhasticheskikh protsessov, Nauka, M., 1977

[2] Azencott R., “Behavior of diffusion semi-groups at infinity”, Bull. Soc. Math. France, 102 (1974), 193–240 | MR | Zbl

[3] Elworthy K. D., Stochastic Differential Equations on Manifolds, London Math. Soc. Lect. Note Ser., 70, Cambridge Univ. Press, Cambridge, 1982 | MR | Zbl

[4] Elworthy K. D., “Stochastic flows and the $C_0$-diffusion property”, Stochastics, 6 (1982), 233–238 | MR | Zbl

[5] Gliklikh Yu. E., A necessary and sufficient condition for completeness of stochastic flows continuous at infinity, Warwick preprint, No 8, August, 2004

[6] Gliklikh Yu. E., Morozova L. A., “Conditions for global existence of solutions of ordinary differential, stochastic differential and parabolic equations”, Internat. J. Math. Math. Sci., 2004, no. 17, 901–912 | DOI | MR | Zbl

[7] Li X.-M., “Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers”, Potential Anal., 3 (1994), 339–357 | DOI | MR | Zbl

[8] Schwartz L., “Processus de Markov et desintégrations régulières”, Ann. Inst. Fourier (Grenoble), 27:3 (1977), 211–277 | MR | Zbl

[9] Schwartz L., “Le semi-groupe d'une diffusion en liaison avec les trajectoires”, Sém. de Probabilités XXIII, Lect. Notes Math., 1372, Springer, Berlin, 1989, 326–342 | MR