Related papers: The geodesic dispersion phenomenon in random field…
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.…
Information geometry is concerned with the application of differential geometry concepts in the study of the parametric spaces of statistical models. When the random variables are independent and identically distributed, the underlying…
Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher…
We develop a geometric framework for irreversible transport phenomena in which macroscopic evolution equations arise from the combined structure of a thermodynamic state metric and an Onsager-based dissipation metric. The construction…
There are many open questions pertaining to the statistical analysis of random objects, which are increasingly encountered. A major challenge is the absence of linear operations in such spaces. A basic statistical task is to quantify…
In this article, we analyze three classes of time-reversal of a Markov process with Gaussian noise on a manifold. We first unveil a commutativity constraint for the most general of these time-reversals to be well defined. Then we give a…
We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natural metric given by the models. We provide the…
A model in statistical mechanics, characterised by the corresponding Gibbs measure, is a subset of the totality of probability distributions on the phase space. The shape of this subset, i.e., the geometry, then plays an important role in…
We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…
The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that…
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…
We develop, simulate and extend an initial proposition by Chaves et al. concerning a random incompressible vector field able to reproduce key ingredients of three-dimensional turbulence in both space and time. In this article, we focus on…
The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. A Gaussian free field is a spin system that takes values in Euclidean space, and this article…
Geometrical model for material Dirac wave field and for Maxwell electromagnetic field is suggested where above fields are considered as propagating regions of the space itself with distorted euclidean geometry. It is shown that equations…
Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from…
This paper analyses the turbulent energy cascade from the perspective of statistical mechanics, and relates inter-scale energy fluxes to statistical irreversibility and information-entropy production. The microscopical reversibility of the…
The appealing connection between non-Euclidean geometries and defects in solids is brought forth in this article. Drawing a correspondence between the nature of a defect and a specific geometric property of the material space not only…
An expression for the joint probability distribution of the principal curvatures at an arbitrary point in the ensemble of isosurfaces defined on isotropic Gaussian random fields on Rn is derived. The result is obtained by deriving symmetry…
Quantum mechanics is sensitive to the geometry of the underlying space. Here, we present a framework for quantum scattering of a non-relativistic particle confined to a two-dimensional space. When the motion manifold hosts localized…
We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Mat\'ern processes on Euclidean space, spheres, manifolds and…