Geodesic
The first problem of diffusion on differentiable manifolds
Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 549-557

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Let $\{\xi_\Delta(k),\ k=0,1,\dots,n=n(\Delta)\}$be a sequence of random walks on a differentiable manifold $M$. In this paper, we obtain the classical conditions for convergence of $\xi_\Delta$ to an inhomogeneous diffusion process $\xi(t)$ in terms of weak convergence of transition probabilities $P_\Delta(t_k,x;t,\Gamma)$ using some modification of Khintchine's idea from [1]. One of many consequences of the result is a limit theorem for convolutions of noncommuting probability measures on Lie groups. 
@article{TVP_1972_17_3_a13,
     author = {G. M. Sobko},
     title = {The first problem of diffusion on differentiable manifolds},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {549--557},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1972_17_3_a13/}
}
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G. M. Sobko. The first problem of diffusion on differentiable manifolds. Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 549-557. http://geodesic.mathdoc.fr/item/TVP_1972_17_3_a13/