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Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds
Abstract
We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace–Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in SU(2) and SL(2,R) equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.
... The work [BBC22] analyses the metric structure, particularly near characteristic points, induced on surfaces embedded in three-dimensional contact sub-Riemannian manifolds, and [BBCH21] introduces and studies properties of a canonical stochastic process on surfaces in three-dimensional contact sub-Riemannian manifolds which exhibits different behaviours near an elliptic characteristic point and a hyperbolic characteristic point. ...
... The present article aims to initiate further studies of hypersurfaces embedded in higher-dimensional contact sub-Riemannian manifolds. We intrinsically construct a sub-Laplacian on hypersurfaces in contact sub-Riemannian manifolds, which for surfaces in three-dimensional contact sub-Riemannian manifolds gives rise to the generator of the stochastic process obtained in [BBCH21] by means of Riemannian approximations, and we use our analysis to propose model cases for this setting. Some notions such as horizontal connectivity, horizontal connection and horizontal mean curvature on hypersurfaces in sub-Riemannian manifolds are studied by Tan and Yang [TY04]. ...
... We then state the result that it emerges as the limit of Laplace-Beltrami operators. This particularly implies that the operator ∆ 0 constructed in [BBCH21] on surfaces in three-dimensional contact sub-Riemannian manifolds coincides with the intrinsic sub-Laplacian defined in this article for the case n = 1. ...
We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace-Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.
... If in the construction of the Laplace-Beltrami operator instead of the Riemannian volume we take any smooth volume, then there is no singularity present (except the degeneracy of the principal symbol) and such operators can be handled using essentially the theory of Hörmander operators even though the singularity still manifests itself in various forms [20,60]. We should also mention separately articles [9,55], where some results concerning analysis of some structures with similar singularities were obtained. In [55] the authors studied the heat content on domains with characteristic points, while in [9] induced stochastic processes on surfaces in the Heisenberg group are studied. ...
... We should also mention separately articles [9,55], where some results concerning analysis of some structures with similar singularities were obtained. In [55] the authors studied the heat content on domains with characteristic points, while in [9] induced stochastic processes on surfaces in the Heisenberg group are studied. ...
The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.
... Gauge balls are actually characterized by such a weighted mean value property for ∆ H n -harmonic functions as proved by Lanconelli in [23]. The metric balls B R defined in (3) are not the unique choice of "balls" adapting to the subRiemannian features of the Heisenberg group. For instance, the Carnot-Carathédory balls play somehow the role of the geodesic balls in H n . ...
... In this section we collect some preliminary material that will be used in the rest of the paper. We shall recall some known notions for the study of smooth hypersurfaces in H n , and we refer the reader to [11,34,32,7,12,35,20,37,10,2,9,3] for several insights and different perspectives and approaches to the geometry of submanifolds in various subRiemannian settings. Being ·, · the metric defined in the Introduction (with induced norm | · |), we denote by ∇ the Levi-Civita connection associated to this metric. ...
In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces.
... Gauge balls are actually characterized by such a weighted mean value property for ∆ H n -harmonic functions as proved by Lanconelli in [23]. The metric balls B R defined in (3) are not the unique choice of "balls" adapting to the subRiemannian features of the Heisenberg group. For instance, the Carnot-Carathédory balls play somehow the role of the geodesic balls in H n . ...
... In this section we collect some preliminary material that will be used in the rest of the paper. We shall recall some known notions for the study of smooth hypersurfaces in H n , and we refer the reader to [11,34,32,7,12,35,20,37,10,2,9,3] for several insights and different perspectives and approaches to the geometry of submanifolds in various subRiemannian settings. Being ·, · the metric defined in the Introduction (with induced norm | · |), we denote by ∇ the Levi-Civita connection associated to this metric. ...
In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces
... We introduce the horizontal Hessian for classifying characteristic points (cf. also [BBCH20]). Fix an affine connection ∇ on the distribution D. Then, the horizontal Hessian of u ∈ C 2 (M ) is the (0, 2)-tensor on D, defined by ...
... As was the case for the horizontal mean curvature, the definition of K 0 restricted at Σ is independent on the choice of the local defining function. The intrinsic Gaussian curvature has been introduced in [BTV17] for the Heisenberg group and in [BBCH20] for the general 3D contact case. We refer to those articles for further details. ...
Sub-Riemannian geometry is a particularly rich class of metric structures, which generalizes Riemannian geometry, where a smoothly varying metric is defined only on a subset of preferred directions of the tangent space at each point of a smooth manifold M (called horizontal directions). Under the so-called Hörmander condition, M is horizontally-path connected, and the usual length-minimization procedure yields a well-defined metric. TheLaplace-Beltrami operator is generalized by the sub-Laplacian which is subelliptic, but has nonetheless suitable regularity properties (in particular, it is hypoelliptic). In this thesis, we investigate the heat content asymptotics and related topics in sub-Riemannian geometry.
... In addition, we show that for "model" surfaces in the model spaces SU (2) and SL(2) (cf. [BBCH21], see also [BH22]), it holds a 2 = a 3 = 0 as for the horizontal plane in H. ...
We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in 10.1016/j.na.2015.05.006 and arXiv:1703.01592v3 for the Heisenberg group.
... If we restrict ourselves to a subsurface Σ ⊆ M , then the picture is quite different. If h ε is the induced metric on Σ from g ε , then d hε does not converge to a metric compatible with the topology if it converges at all, see [3,4] for details. Seeing that this limit breaks the topology of Σ, it is interesting to study the limit of the Gauss-Bonnet formula with respect to h ε as ε ↓ 0. See [5,7] for previous results relating topology to sub-Riemannian invariants. ...
We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider 3 dimensional contact sub-Riemannian manifolds with a corresponding metric extension and obtain a pure sub-Riemannian metric in the limit for subsurfaces. In particular, we are able to recover topological information concentrated around the characteristic set of points where the tangent space to the surface and contact structure coincide.
... We introduce the horizontal Hessian for classifying characteristic points (cf. also [BBCH20]). Fix an affine connection∇ on the distribution D. Then, the horizontal Hessian of u ∈ C 2 (M ) is the (0, 2)-tensor on D, defined as ...
We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this note is the introduction of a concept of mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence we partially answer to a question posed by Danielli-Garofalo-Nhieu in [Danielli D., Garofalo N., Nhieu D.M., Proc. Amer. Math. Soc., 2012], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.
We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-defined at singular points of the boundary distribution, the appearance of gliding and creeping orbits, and the behavior of reflections at wavefronts.
We then study some concrete tables in \begin{document}$ 3 $\end{document}-dimensional euclidean space endowed with the standard contact structure. These can be interpreted as planar magnetic billiards, of varying magnetic strength, for which the magnetic strength may change under reflection. For each table we provide various results regarding periodic trajectories, gliding orbits, and creeping orbits.
Given a surface S in a 3D contact sub-Riemannian manifold M , we investigate the metric structure induced on S by M , in the sense of length spaces. First, we define a coefficient [[EQUATION]] at characteristic points that determines locally the characteristic foliation of S . Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this paper is the introduction of a concept of a mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence, we partially answer to a question posed by Danielli, Garofalo and Nhieu in [D. Danielli, N. Garofalo and D. M. Nhieu, Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group, Proc. Amer. Math. Soc. 140 2012, 3, 811–821], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.
Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory.
We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean \(C^{2}\)-smooth surface in the Heisenberg group \(\mathbb {H}\) away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean \(C^{2}\)-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in \(\mathbb {H}\) is provided.
We study the evolution of the heat and of a free quantum particle (described by the Schrödinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric ds2=dx2+|x|-2αdθ2, where x∈R, θ∈T and the parameter α∈R. For α≤-1 this metric describes cone-like manifolds (for α=-1 it is a flat cone). For α=0 it is a cylinder. For α≥1 it is a Grushin-like metric. We show that the Laplace-Beltrami operator δ is essentially self-adjoint if and only if α∉(-3,1). In this case the only self-adjoint extension is the Friedrichs extension δF, that does not allow communication through the singular set (x=0) both for the heat and for a quantum particle. For α∈(-3,-1] we show that for the Schrödinger equation only the average on θ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is δF) cannot. For α∈(-1,1) we prove that there exists a canonical self-adjoint extension δB, called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the L1 norm for the heat equation) of the Markovian extensions δF and δB, proving that δF is stochastically complete at the singularity if and only if α≤-1, while δB is always stochastically complete at the singularity.
The problem of the local summability of the sub-Riemannian mean curvature H of a hypersurface M in the Heisenberg group, or in more general Carnot groups, near the characteristic set of M arises naturally in sev-eral questions in geometric measure theory. We construct an example which shows that the sub-Riemannian mean curvature H of a C 2 surface M in the Heisenberg group H 1 in general fails to be integrable with respect to the Rie-mannian volume on M .
The characteristic set C(S) of a codimension 1 submanifold S in the Heisenberg group ℍn consists of those points where the tangent space of S coincides with the space spanned by the left invariant horizontal vector fields of ℍn. We prove that if S is C1 smooth then C(S) has vanishing 2n + 1-dimensional Hausdorff measure with respect to the Heisenberg metric. If S is C2 smooth then C(S) has Hausdorff dimension less or equal than n, both with respect to the Euclidean and Heisenberg metrics. On the other hand, C(S) can have a positive 2n-dimensional Hausdorff measure with respect to the Euclidean metric even for hypersurfaces of class ∩ C1,x. This is shown by constructing a function of class ∩ C1,x with a prescribed gradient on a large measure set.
In this paper we study the Carnot-Caratheodory metrics on SU(2) ≃ S3, SO(3) and SL(2) induced by their Cartan decomposition and by the Killing form. Besides computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on SU(2) projects on the so called lens spacesL(p,q). Also for lens spaces, we compute the cut loci (globally). For SU(2) the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group.
In this paper we study the small time asymptotics for the heat kernel on a
sub-Riemannian manifold, using a perturbative approach. We then explicitly
compute, in the case of a 3D contact structure, the first two coefficients of
the small time asymptotics expansion of the heat kernel on the diagonal,
expressing them in terms of the two basic functional invariants $\chi$ and
$\kappa$ defined on a 3D contact structure.
We investigate distributions of hyperbolic Bessel processes. We find links
between the hyperbolic cosinus of the hyperbolic Bessel processes and the
functionals of geometric Brownian motion. We present an explicit formula of
Laplace transform of hyperbolic cosinus of hyperbolic Bessel processes and some
interesting different probabilistic representations of this Laplace transform.
We express the one-dimensional distribution of hyperbolic Bessel process in
terms of other, known and independent processes. We present some applications
including a new proof of Bougerol's identity and it's generalization. We
characterize the distribution of the process being hyperbolic sinus of
hyperbolic Bessel processes.
We give a complete classification of left-invariant sub-Riemannian structures
on three dimensional Lie groups in terms of the basic differential invariants.
As a corollary we explicitly find a sub-Riemannian isometry between the
nonisomorphic Lie groups SL(2) and $A^+(\R)\times S^1$, where $A^+(\R)$ denotes
the group of orientation preserving affine maps on the real line.
We introduce a notion of geodesic curvature k ζ k_{\zeta} for a smooth horizontal curve 𝜁 in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve
d SR 2 ( ζ ( t ) , ζ ( t + ε ) ) = ε 2 - k ζ 2 ( t ) 720 ε 6 + o ( ε 6 ) . d_{\mathrm{SR}}^{2}(\zeta(t),\zeta(t+\varepsilon))=\varepsilon^{2}-\frac{k_{\zeta}^{2}(t)}{720}\varepsilon^{6}+o(\varepsilon^{6}).
The sub-Riemannian distance is not smooth on the diagonal; hence the result contains the existence of such an asymptotics. This can be seen as a higher-order differentiability property of the sub-Riemannian distance along smooth horizontal curves. It generalizes the previously known results on the Heisenberg group.
This book explains the foundations of holomorphic curve theory in contact geometry. By using a particular geometric problem as a starting point the authors guide the reader into the subject. As such it ideally serves as preparation and as entry point for a deeper study of the analysis underlying symplectic field theory.
An introductory chapter sets the stage explaining some of the basic notions of contact geometry and the role of holomorphic curves in the field. The authors proceed to the heart of the material providing a detailed exposition about finite energy planes and periodic orbits (chapter 4) to disk filling methods and applications (chapter 9).The material is self-contained. It includes a number of technical appendices giving the geometric analysis foundations for the main results, so that one may easily follow the discussion. Graduate students as well as researchers who want to learn the basics of this fast developing theory will highly appreciate this accessible approach taken by the authors.
We present a notion of geodesic curvature for smooth horizontal curves in a contact sub-Riemannian manifold, measuring how far a horizontal curve is from being a geodesic. This geodesic curvature consists in two functions that both vanish along a smooth horizontal curve if and only if this curve is a geodesic. The main result of this thesis is the metric interpretation of these geodesic curvature functions. This interpretation consists in seeing the geodesic curvature functions as the first corrective coefficients in the Taylor expansion of the sub-Riemannian distance between two close points on the curve.
We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh’s Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh’s Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).
Le texte est consacre a l'etude des structures de contact qui sont invariantes par le gradient d'une fonction de morse. On donne en particulier, en dimension 3, une condition topologique necessaire et suffisante pour l'existence de telles structures. Pour cela, on explique comment utiliser les champs de vecteurs qui preservent une structure de contact pour decrire le feuilletage caracteristique des surfaces et ses deformations au cours de certaines isotopies
This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.
We study relations between particular known diffusions and special functions. A new class of diffusions related to hypergeometric functions is defined. A very interesting particular case of this class consists of hyperbolic Bessel processes. Bibliography: 6 titles.
The second volume concentrates on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. These subjects are made accessible in the many concrete examples that illustrate techniques of calculation, and in the treatment of all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appear for the first time in this book. © John Wiley & Sons Ltd 1979, 1994 and Cambridge University Press 2000.
This article addresses the mathematical foundations of rhumblines or loxodrome curves. These curves are critical to navigation and small-scale charting by virtue of the fact that they provide an efficient routeing from one point on a surface to another by means of a constant ‘course angle‘. This article will develop the necessary mathematical relations for the construction of such a curve, then apply the relations to both spherical and oblate-spheroidal surfaces. The purpose of this article is to produce a superior oblate-spheroidal loxodrome curve, which better models curves or routes of constant course on the actual, approximately oblatespheroidal, Earth.
We define hyperbolic von Mises distributions in any integral dimension as exit distributions of hyperbolic Brownian motion (H<sup>(α)</sup> <sub>t</sub>,t≥0) with drift outside hyperbolic balls centred on the starting point H<sub>0</sub> . Bidimensional unwrapped hyperbolic distributions are also considered.
Libro de probabilidad. Contenido: Introducción; Martingalas; Procesos de Markov; Integración estocástica; Representación de martingalas; Hora media local; Generadores y tiempo de inversión; Teorema de Girsanov y aplicación; Ecuaciones diferenciales estocásticas; Funciones aditivas de movimiento browniano; Procesos de Bessel y teorema Ray-Knight; Recorrido; Teorema de límite y distribución.
The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) and results from the combination of two techniques: surgery, which produces many contact structures, and tomography, which allows one to analyse a contact structure given a priori and to create from it a combinatorial image. The surgery methods are based on a theorem of Y. Eliashberg -- revisited by R. Gompf [math.GT/9803019] -- and produces holomorphically fillable contact structures on closed manifolds. Tomography theory, developed in parts 2 and 3, draws on notions introduced by the author and yields a small number of possible models for contact structures on each of the manifolds listed above. Comment: Abstract added in migration
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