Hypoelliptic Laplacian Research Articles

Hypoelliptic Laplacian and twisted trace formula

Hypoelliptic Laplacian and twisted trace formula

Parabolic Anderson model on Heisenberg groups: The Itô setting

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as ∂tu=12Δu+uW˙α, where Δ is the hypoelliptic Laplacian on Hn and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as α>n2; (iii) Give some basic moment estimates for the solution u(t,x).

The Shifted Harmonic Oscillator and the Hypoelliptic Laplacian on the Circle

We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half-plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This relies on a novel integral formula for the spectral projections which also gives asymptotics for Laguerre polynomials in a large-parameter regime.

Eta invariants and the hypoelliptic Laplacian

The purpose of this paper is to give a new proof of the results of Moscovici and Stanton on orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals that uses the hypoelliptic Laplacian with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct a suitable Itô calculus associated with certain hypoelliptic diffusions.

Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau, and Shen)

Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau, and Shen)

Hypoelliptic Laplacian and twisted trace formula

In this Note, we give an explicit geometric formula for twisted orbital integrals using the method of the hypoelliptic Laplacian developed by Bismut. We apply this formula to evaluate the leading term in the asymptotic expansion of the equivariant Ray–Singer analytic torsion on compact locally symmetric spaces.

Coupling in the Heisenberg group and its applications to gradient estimates

We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally, we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.

Boundary Conditions and Subelliptic Estimates for Geometric Kramers-Fokker-Planck Operators on Manifolds with Boundaries

This article is concerned with maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic Laplacian.

Toeplitz Operators, Analytic Torsion, and the Hypoelliptic Laplacian

The purpose of this paper is to explain how Toeplitz operators can be used in studying asymptotic torsion, and also in the theory of the hypoelliptic Laplacian. The role of the hypoelliptic Laplacian in the explicit computation of orbital integrals will be described. The geodesic flow will be viewed as implementing a dynamical version of Fourier transform.

Hypoelliptic Laplacian and probability

The purpose of this paper is to describe the probabilistic aspects underlying the theory of the hypoelliptic Laplacian, as a deformation of the standard elliptic Laplacian. The corresponding diffusion on the total space of the tangent bundle of a Riemannian manifold is a geometric Langevin process, that interpolates between the geometric Brownian motion and the geodesic flow. Connections with the central limit theorem for the occupation measure by the geometric Brownian motion are emphasized. Spectral aspects of the hypoelliptic deformation are also provided on tori. The relevant hypoelliptic deformation of the Laplacian in the case of Riemann surfaces of constant negative curvature is briefly described, in connection with Selberg's trace formula.

The hypoelliptic Laplacian, analytic torsion and Cheeger–Müller Theorem

The purpose of this Note is to prove a formula relating the hypoelliptic Ray–Singer metric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.

Short-time asymptotics of heat kernels for a class of hypoelliptic operators

We compute the short-time asymptotics of heat kernels for a family of hypoelliptic operators, and we relate these to the value function of an associated variational problem previously investigated in the control theory literature. These heat kernels generalize to the noncontact case a heat kernel that has been previously obtained for the contact hypoelliptic Laplacian defined on the Heisenberg group.

Laplacien hypoelliptique et intégrales orbitales

We give a new approach to orbital integrals based on the hypoelliptic Laplacian. The formalism unifies the Atiyah–Singer index theorem and the trace formula. To cite this article: J.-M. Bismut, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace–Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU ( 2 ) , SO ( 3 ) , SL ( 2 ) (with the metrics inherited by the Killing form), and the group SE ( 2 ) of rototranslations of the plane.

The hypoelliptic Laplacian on a compact Lie group

Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G × g , which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula.

Loop Spaces and the hypoelliptic Laplacian

AbstractThe purpose of this paper is to introduce some ideas that motivated the construction of the hypoelliptic Laplacian as a deformation of Hodge theory, which interpolates between Hodge theory and the geodesic flow. Results obtained with Lebeau on the analysis of the hypoelliptic Laplacian are also presented. © 2007 Wiley Periodicals, Inc.

Equations de Fokker-Planck géométriques II : estimations hypoelliptiques maximales

We study some analytic properties of the hypoelliptic Laplacian of Jean-Michel Bismut, and more generally, of geometric Fokker-Planck operators P acting on the cotangent bundle Σ=T * X of a compact Riemannian manifold X. In particular, we prove a maximal hypoelliptic estimate for P, and we get bounds on the location of the spectrum of P.

Laplacien hypoelliptique et torsion analytique

We establish analytical properties of the hypoelliptic Laplacian on the cotangent bundle of a Riemannian manifold. We show that it is, in the proper sense, a deformation of the classical Laplacian on X. We give a formula relating the analytic torsion of the hypoelliptic Laplacian to the Ray–Singer analytic torsion of the Laplacian of X. To cite this article: J.-M. Bismut, G. Lebeau, C. R. Acad. Sci. Paris, Ser. I 341 (2005).