Stochastic sub-Riemannian geodesics on the Grushin distribution (original) (raw)
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arXiv: Probability, 2020
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We analyze the disordered geometry resulting from random permutations of Euclidean space. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment.
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Geodesics of Random Riemannian Metrics I: Random Perturbations of Euclidean Geometry
arXiv (Cornell University), 2012
We analyze the disordered geometry resulting from random permutations of Euclidean space. 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 perturbations, and 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. In Part II, we will use this to prove that a geodesic with random initial conditions is almost surely not minimizing. We also develop in this paper some general results on conditional Gaussian measures.
Geodesics of Random Riemannian Metrics II: Minimizing Geodesics
arXiv (Cornell University), 2012
We continue our analysis of geodesics in quenched, random Riemannian environments. In this article, we prove that a geodesic with randomly chosen initial conditions is almost surely not minimizing. To do 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.
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Cornell University - arXiv, 2021
Random fields are mathematical structures used to model the spatial interaction of random variables along time, with applications ranging from statistical physics and thermodynamics to system's biology and the simulation of complex systems. Despite being studied since the 19th century, little is known about how the dynamics of random fields are related to the geometric properties of their parametric spaces. For example, how can we quantify the similarity between two random fields operating in different regimes using an intrinsic measure? In this paper, we propose a numerical method for the computation of geodesic distances in Gaussian random field manifolds. First, we derive the metric tensor of the underlying parametric space (the 3 × 3 first-order Fisher information matrix), then we derive the 27 Christoffel symbols required in the definition of the system of non-linear differential equations whose solution is a geodesic curve starting at the initial conditions. The fourth-order Runge-Kutta method is applied to numerically solve the non-linear system through an iterative approach. The obtained results show that the proposed method can estimate the geodesic distances for several different initial conditions. Besides, the results reveal an interesting pattern: in several cases, the geodesic curve obtained by reversing the system of differential equations in time does not match the original curve, suggesting the existence of irreversible geometric deformations in the trajectory of a moving reference traveling along a geodesic curve.
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Radial processes for sub-Riemannian Brownian motions and applications
Electronic Journal of Probability
We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng's type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.