11 research outputs found
Geodesics of Random Riemannian Metrics
We analyze the disordered Riemannian geometry resulting from random
perturbations of the Euclidean metric. We focus on geodesics, the paths traced
out by a particle traveling in this quenched random environment. By taking the
point of the view of the particle, we show that the law of its observed
environment is absolutely continuous with respect to the law of the random
metric, and we provide an explicit form for its Radon-Nikodym derivative. We
use this result to prove a "local Markov property" along an unbounded geodesic,
demonstrating that it eventually encounters any type of geometric phenomenon.
We also develop in this paper some general results on conditional Gaussian
measures. Our Main Theorem states that a geodesic chosen with random initial
conditions (chosen independently of the metric) is almost surely not
minimizing. To demonstrate this, we show that a minimizing geodesic is
guaranteed to eventually pass over a certain "bump surface," which locally has
constant positive curvature. By using Jacobi fields, we show that this is
sufficient to destabilize the minimizing property.Comment: 55 pages. Supplementary material at arXiv:1206.494
An optimal series expansion of the multiparameter fractional Brownian motion
We derive a series expansion for the multiparameter fractional Brownian
motion. The derived expansion is proven to be rate optimal.Comment: 21 pages, no figures, final version, to appear in Journal of
Theoretical Probabilit