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Random perturbation to the geodesic equation. (English) Zbl 1372.60083

The paper deals with a geodesic equation which is represented by a canonical differential equation on the orthonormal frame bundle. Now, consider the randomly perturbed equation and let \(n\) be the dimension of the state space. It is shown that after some rescaling its limit is a Brownian motion scaled by \(\frac{8}{n(n-1)}\). A further result is the convergence of the horizontal lifts to the orthonormal frame bundle. The limit in this case is also a scaled Brownian motion.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
58J65 Diffusion processes and stochastic analysis on manifolds
37H10 Generation, random and stochastic difference and differential equations
53B05 Linear and affine connections
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References:

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