Geodesic
Diffusion processes fnd Riemannian geometry
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 30 (1975) no. 1, pp. 1-63

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This paper studies the asymptotic behaviour as $t\to 0$ of the transition density of a diffusion process on a smooth manifold. The results are stated in the language of differential geometry. We look at applications of the asymptotic formulae to the local structure of diffusion processes and to spectral theory. 
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     author = {S. A. Molchanov},
     title = {Diffusion processes fnd {Riemannian} geometry},
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S. A. Molchanov. Diffusion processes fnd Riemannian geometry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 30 (1975) no. 1, pp. 1-63. http://geodesic.mathdoc.fr/item/RM_1975_30_1_a0/