Geodesic
Geodesies with random curvature on Riemannian and pseudo-Riemannian manifolds
Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 24 (2003), pp. 99-106

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We introduce a notion of renewing geodesic whose curvature is a random process. We demonstrate that the norm of Jacobi field along this geodesic line is of exponential growth, and that there exist infinitely many conjugate points with probability 1. Also we find the upper bound for the average distance between conjugate points. 
@article{KUTGS_2003_24_a9,
     author = {V. G. Lamburt and \`E. R. Rozendorn and D. D. Sokolov and V. N. Tutubalin},
     title = {Geodesies with random curvature on {Riemannian} and {pseudo-Riemannian} manifolds},
     journal = {Trudy Geometricheskogo Seminara},
     pages = {99--106},
     publisher = {mathdoc},
     volume = {24},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/KUTGS_2003_24_a9/}
}
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%J Trudy Geometricheskogo Seminara
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V. G. Lamburt; È. R. Rozendorn; D. D. Sokolov; V. N. Tutubalin. Geodesies with random curvature on Riemannian and pseudo-Riemannian manifolds. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 24 (2003), pp. 99-106. http://geodesic.mathdoc.fr/item/KUTGS_2003_24_a9/