Geodesics of Random Riemannian Metrics (original) (raw)
Related papers
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.
Geodesics of Random Riemannian Metrics: Supplementary Material
This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem from [LW10], specialized to the setting of this article. In Appendix C, we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix D, we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix E, we present some analytic estimates which we use in the article. In Appendix F, we present the construction of the conditional mean operator for Gaussian measures. In Appendix G, we describe Fermi normal coordinates, which we use in our construction of the bump metric.
Geodesic curves in Gaussian random field manifolds
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.
arXiv: Probability, 2020
We describe, in an intrinsic way and using the global chart provided by Ito's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These stochastic processes are no longer smooth paths but they are still critical points of a regularised stochastic energy functional. We consider stochastic geodesics on compact Riemannian manifolds and also on (possibly infinite dimensional) Lie groups. Finally the question of existence of such stochastic geodesics is discussed: we show how it can be approached via forward-backward stochastic differential equations.
Stochastic sub-Riemannian geodesics on the Grushin distribution
2014
Recent years have seen intensive scientiflc activities of describ- ing difiusion processes with Brownian covariance given by a Riemannian metric on a manifold. In our paper the dynamics is specifled through a stochastic variational principle for a generalization of the classical action, with a given kinetic Riemannian metric. In short, we introduce the concept of stochastic sub-Riemannian geodesics and flnd their equations in the case of Grushin distribution. We also discuss the number of stochastic geodesics between any two given points and calculate their energies.
Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
ESAIM: Control, Optimisation and Calculus of Variations
We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.
A Stochastic Look at Geodesics on the Sphere
Lecture Notes in Computer Science, 2017
We describe a method allowing to deform stochastically the completely integrable (deterministic) system of geodesics on the sphere S 2 in a way preserving all its symmetries.