arXiv research

We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive…

Study solves $\infty(x)$-equation in Grushin spaces using adapted jets.

Study sub-Finsler geometry in time-optimal control for specific distributions.

Computed distortion coefficients for the α-Grushin plane.

Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.

Study on quantum particle evolution on Grushin cylinder, embedding in R^3.

Bi-Lipschitz embedding of a generalized Grushin plane in Euclidean space.

New cylindrical solutions found for Grushin-type problem.

Research finds double bubbles in Grushin plane with specific geometric properties.

The study examines optimal synthesis in a radially symmetric Grushin space with conditions on the weight function.

Fundamental solutions found for p-Laplace equations in Heisenberg and Grushin spaces.

Study geodesics on Grushin spaces, proving upper bounds on conjugate times.

Study geometric quantum confinement on special incomplete Riemannian manifolds.

New sub-Riemannian spaces with boundary meet curvature-dimension condition.

The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.

Classifies quantum particle behavior on a special cylinder.

This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y…

Study shows failure of curvature-dimension conditions on sub-Riemannian manifolds.

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometric…

Counterexample disproves gluing theorem for MCP metric spaces.

Study on heat flow across two half-lines with special boundary conditions.

The study proves sub-Riemannian manifolds cannot satisfy $\mathrm{CD}$ conditions unless they are Riemannian.

We study Monge's optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnorma…

Extensions of Brownian motion to singular surfaces are studied.

New Weyl's laws discovered for compact spaces with Ricci curvature bounds.

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.

Paper studies viscosity solutions in unique Martinet spaces.

New theorem on embedding Moebius bands in 3D space.

Researchers solve p-Laplace equation in Hörmander vector fields.

Generalizes Cheeger inequality to Carnot-Carathéodory spaces.

New trigonometric method for convex sets aids control problems.

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…

Ideal sub-Riemannian manifolds support interpolation inequalities for optimal transport.

The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.

Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.

For a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\ all pieces…

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $μ$, …

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…

Quantitative estimates for inequalities on sub-Riemannian manifolds.

The paper explores solutions to the distributional Bellman equation in reinforcement learning.

Proposes vMF distribution for skewed elliptical distributions.

This paper examines how the choice of prior distribution affects likelihoods of out-of-distribution inputs in deep generative models.

Study calculates tail risk for various mixture distributions.

No stable distributions found in financial data, study shows.

New constrained mixtures create asymmetric distributions for better modeling.

New wealth distribution model based on $κ$-deformation of Gamma distribution.

This study suggests replacing Ising distribution with Cox distribution.

New algorithms for learning under s-concave distributions, including Pareto and t-distributions.