Random perturbation to the geodesic equation

Li, Xue-Mei ^[1]
1. [1] University of Warwick
  
  University of Warwick
  
  Reino Unido
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 1, 2016, págs. 544-566
Idioma: inglés
DOI: 10.1214/14-AOP981
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Resumen
- We study random “perturbation” to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm 1. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by 8n(n−1) where n is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.