Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.
problem Large deviations for hypoelliptic diffusion measures on sub-Riemannian manifolds.
method Rough path theory and manifold-valued Malliavin calculus.
result Proved a large deviation principle for pinned hypoelliptic diffusion measures.
Study small-time fluctuations for sub-Riemannian diffusion loops.
problem Analyzing fluctuations of sub-Riemannian diffusion processes.
method Analyzes small-time fluctuations of diffusion processes with sub-Riemannian structure, identifying degenerate and non-degenerate covariance matrices.
result Rescaled fluctuations converge to a non-degenerate limiting diffusion loop.
New method for sampling diffusion bridges on sub-Riemannian manifolds.
problem Sampling conditioned diffusion processes on sub-Riemannian manifolds is challenging.
method Score matching for machine learning, adapted to non-holonomic frames.
result Demonstrated method works on Heisenberg group and other sub-Riemannian manifolds.
Study on diffusion in non-complete sub-Riemannian manifolds with specific conditions.
problem Analyzing diffusion in incomplete sub-Riemannian manifolds.
method Identifying conditions for Gaussian-type upper bounds and logarithmic asymptotics of heat kernels.
result Optimal constant in exponent for Gaussian-type upper bounds and concentration of diffusion bridge measures.
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-R…
Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.
problem Developing stochastic processes on sub-Riemannian manifolds.
method Introduces stochastic development using Cartan connections, derives generator, and provides conditions for existence.
result Derives a general expression for the generator of the stochastic process and provides conditions for the existence of a Cartan connection.
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the…
A new image completion method inspired by brain cells.
problem Image restoration from corrupted data.
method Biologically-inspired sub-Riemannian model with frequency and phase.
result Completion of two-dimensional images using cortical cell responses.
Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.
problem Analyzing sub-Riemannian heat kernels and their derivatives on incomplete manifolds.
method Localized asymptotic analysis, focusing on minimizing geodesics and the non-abnormal cut locus.
result Uniform bounds and expansions for heat kernels and their derivatives on compacts, including the diffusion bridge measure.
Two stochastic representations help prove infinite lifetime and bound gradients on sub-Riemannian manifolds.
problem Proving infinite lifetime and finding gradient bounds on sub-Riemannian manifolds.
method Using stochastic representations of heat semigroups and interplay between sub-Riemannian geometry and connections.
result Explicit pointwise bounds for horizontal gradients on Carnot groups.
In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As …
We relate some basic 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 at…
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…
We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps) focus on mining massive data sets using w…
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…
Researchers compute heat kernel coefficients for 2D diffusion operators.
problem Analyzing heat kernel coefficients for 2D hypoelliptic operators.
method Explicit computation of heat kernel coefficients and interpretation in terms of curvature.
result Interpretation of heat kernel asymptotics for non-sub-Riemannian operators.
Study on removing sets and uniqueness of diffusion operators on various spaces.
problem Determining the effect of removing small sets on the self-adjointness and uniqueness of diffusion operators.
method Analyzes symmetric diffusion operators on metric measure spaces, proving a truncation result for potentials.
result Characterizes the critical size of removed sets and their effect on operator properties.
Study of sub-Riemannian cubics in SU(2) for long-term dynamics.
problem Optimizing curves in a sub-Riemannian manifold with constraints.
method Analysis of sub-Riemannian Lie quadratics in SU(2).
result Characterization of sub-Riemannian Lie quadratics in SU(2).
The authors first in this paper define a semi-symmetric metric non-holonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation $\nabla…
Study sub-Riemannian geometry on parallelizable distributions.
problem Investigate the geometry of sub-Riemannian structures on non-integrable parallelizable distributions.
method Construct and analyze two linear connections: Weitzenböck and sub-Riemannian.
result Two concrete examples (S^3 and S^7) illustrate the application of sub-Riemannian geometry.
Study optimal transport in 4D sub-Riemannian spaces with many singular geodesics.
problem Existence and uniqueness of optimal transport maps in sub-Riemannian structures.
method Analysis of Monge optimal transport problem in sub-Riemannian manifolds.
result Extension of previous results to sub-Riemannian structures of rank two in 4D.
Study compares H-type sub-Riemannian manifolds using uniform metrics.
problem Comparing H-type sub-Riemannian manifolds with Riemannian metrics.
method Establishes sub-Hessian and sub-Laplacian comparison theorems for a family of approximating Riemannian metrics.
result Proves a sharp sub-Riemannian Bonnet-Myers theorem.
Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.
problem Understanding sub-Riemannian distances on Sasakian manifolds.
method Parallel and mirror maps along geodesics of a taming Riemannian metric.
result Limits of transport maps outside sub-Riemannian cut-locus provide bounds on sub-Riemannian distance.
Study calculates volume of small sub-Riemannian balls in 3D contact manifolds.
problem Computing the volume of small sub-Riemannian balls in 3D contact manifolds.
method Asymptotic expansion and geometric invariants of the sub-Riemannian structure.
result Expressed first geometric coefficients in terms of sub-Riemannian structure invariants.
Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
Note shows Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
problem Proving Cheng-Yau gradient estimate for specific geometric structures.
method Utilizing previous results on sub-Riemannian manifolds and Carnot groups.
result Established Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian
problem Sub-Riemannian geometry and eigenvalues of sub-Laplacian
method Embedding manifold into Hilbert space using eigenfunctions
result Defined spectral distance between sub-Riemannian manifolds
Constant curvature models in sub-Riemannian geometry are explored.
problem Understanding constant curvature models in sub-Riemannian geometry.
method Cohomological description of principal invariants for parabolic geometries.
result Constant curvature models are discussed for specific sub-Riemannian geometries.
We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen \cite{Hughen}. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariant…
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a C2 but not C3 length-minimizer example. result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly sub-Riemannian manifolds and study the geometric properties of such a connection, and obtain the natures of horizontal curvature tensors between horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors further investigate …
Proves C1 regularity for abnormal minimizers in rank 2 sub-Riemannian structures.
problem Regularity of abnormal minimizers in sub-Riemannian structures.
method Proves C1 regularity using length-minimizers in rank 2 sub-Riemannian structures. result All length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class C1. Study sub-Riemannian metrics on compact Lie groups, finding same shortest loops.
problem Exploring sub-Riemannian length spectra on compact Lie groups.
method Restricting Killing form to root spaces of compact Lie groups.
result Same shortest loops in Riemannian and sub-Riemannian cases.
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…
The study explores different definitions of geodesics in sub-Riemannian geometry.
problem Understanding geodesics in sub-Riemannian geometry and their equivalence.
method Review of three variational definitions of geodesics and three definitions of straightest curves.
result Shortest geodesics coincide with straightest geodesics in some sub-Riemannian manifolds.
Study compares sub-Riemannian curvature to optimal control variational problems.
problem Comparing sub-Riemannian curvature to optimal control variational problems.
method Introducing sub-Riemannian Bakry-Émery curvature and proving sub-Laplacian comparison theorems.
result Established sharp measure contraction property for 3-Sasakian manifolds.
New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.
problem Finding complete sub-Riemannian structures satisfying the Minimizing Sard conjecture.
method Techniques from nonsmooth analysis and geometric measure theory.
result Complete sub-Riemannian structures associated with distributions of co-rank 2 or generic distributions of rank ≥ 2 satisfy the Minimizing Sard conjecture.
Proves measure contraction for specific sub-Riemannian structures.
problem Measure contraction properties in sub-Riemannian structures.
method Analytic sub-Riemannian structures and Lipschitz Carnot groups.
result Proves measure contraction properties for the structures.
Introduces canonical connections for sub-Riemannian manifolds with constant symbol.
problem Equivalence problem in sub-Riemannian geometry.
method Introduces canonical grading and compatible affine connection.
result Completely computed structures for contact manifolds of constant symbol.
The paper develops a non-transitive Cartan connection for sub-Riemannian manifolds.
problem Analyzing sub-Riemannian structures and their symmetries.
method Using Lie groupoids and non-transitive Cartan connections.
result A non-transitive Cartan connection is constructed for sub-Riemannian manifolds.
New integrable structures found with abnormal geodesics.
problem Integrable homogeneous sub-Riemannian structures with abnormal geodesics.
method Analysis of equivalence problem for sub-Riemannian Engel structures.
result First known family of examples of integrable homogeneous sub-Riemannian structures with strictly abnormal geodesics.
This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.
problem Classifying singularities of sub-Riemannian geodesics.
method Complete local classification using Legendre fibrations.
result Legendre singularities are completely classified for sub-Riemannian geodesics.
The author calculates curvatures for homogeneous sub-Riemannian manifolds using specific riggings.
problem Calculating curvatures for homogeneous sub-Riemannian manifolds.
method Using special riggings of invariant completely non-holonomic distributions, the author calculates Solov'ev sectional and Ricci curvatures.
result The method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds with a submetry onto a Riemannian manifold.
Study sub-Riemannian surfaces in contact manifolds, proving a Gauss-Bonnet theorem.
problem Proving a Gauss-Bonnet theorem for sub-Riemannian surfaces in contact manifolds.
method Using a family of taming Riemannian metrics, the theorem is derived in the limit.
result Recover topological information of surfaces from geometry around characteristic set.
We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Rieman…
In this letter we exhibit the relation between the isometries of a Riemannian contraction of a sub-Riemannian manifold and those of the sub-Riemannian metric, for to use this relation with two goals: establishing a result about the existence of fixed points of isometries groups; and the other, defining a Multiresolutio…