On smooth solutions of a class of almost hypoelliptic equations

For a class of non-degenerate (regular) almost hypoelliptic equations, which are partially hypoelliptic with respect to certain variables, infinitely differentiable in a specific strip solutions are extracted.     

On integrable with a weight solutions of a class of almost hypoelliptic equations

We prove that the solutions (from the weighted space L_2,δ) of almost hypoelliptic equations belong to the Gevrey classes.     

Anisotropic hypoelliptic estimates for Landau-type operators

We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various linear inhomogeneous kinetic equations, we establish for linear Landau-type operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion.     

Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.     

T-product of hypoelliptic operators

Asymptotics as t0 for the solution of the Cauchy problem for hypoelliptic equations are obtained. Using these results, the existence of the T-product for hypoelliptic operators is proved as is a theorem on the removal of autonomous brackets in the T-product.Translated from Itogi i Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 137–197, 1977.     

Lower bounds for densities of Asian type stochastic differential equations

We obtain lower bounds for densities of solutions of certain hypoelliptic two-dimensional stochastic differential equations where one of the components is the Lebesgue integral of the other. These results are non-trivial extensions of previous work of the authors. In particular, these type of equations are linked to the so-called Asian option set-up.     

On a class of partially hypoelliptic operators in weighted Sobolev spaces

The set of smooth solutions of a class of regular, partially hypoelliptic with respect to the plane x ₁ = 0 equations in a rather wide strip is extracted.     

On a class of almost hypoelliptic polynomials

The paper gives criteria for almost hypoelliptic polynomials P of two and three variables, providing that P(ξ) → ∞ as |ξ| → +∞.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

On almost hypoelliptic polynomials increasing at infinity

It is proved that a polynomial (the symbol of a differential operator), the Newton polygon of which is a rectangular parallelepiped with a vertex at the origin, is almost hypoelliptic if and only if it is regular. Also some algebraic conditions of almost hypoellipticity are obtained for nonregular polynomials increasing at infinity. The results are unimprovable for polynomials of two variables.     

On Solvability of Regular Hypoelliptic Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper the unique solvability of regular hypoelliptic equations in multianisotropic weighted functional spaces is proved by means of special integral representation of functions through a regular operator. The existence of the solutions is proved by constructing approximate solutions using multianisotropic integral operators.     

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-scale homogenization of uniformly parabolic fully nonlinear PDEs.     

Approximations for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.     

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).     

Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations

We prove results on the propagation of Gevrey and analytic wave front sets for a class of hypoelliptic equations with double characteristics.

     

Hypoelliptic Jacobi Convolution Operators on Schwartz Distributions

In this paper we characterize the hypoellipticity of Jacobi convolution operators on Schwartz distributions. In the proof of the main result of this paper the positivity ofthe convolution structure for the inverse of the Jacobi transform plays an essential role. We also study hypoelliptic convolution equations on Chébli-Triméche hypergroups.Mathematics Subject Classification 2000: 46F12     

Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups

We consider the problem of computing heat kernel small-time asymptotics for hypoelliptic Laplacians associated to left-invariant sub-Riemannian structures on unimodular Lie groups of type I. We use the non-commutative Fourier transform of the Lie group together with perturbation theory for semigroups of operators in deriving these asymptotics. We illustrate our approach on the example of the Heisenberg group, and, as an application, we compute the short-time behaviour of the hypoelliptic heat kernel on the step 3 nilpotent Cartan and Engel groups, for which no closed-form expression for the hypoelliptic heat kernel is yet known.     

Hypoelliptic Dunkl Convolution Equations in the Space of Distributions on {\Bbb R}^d

In this article we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.     

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.     

Loop Spaces and the hypoelliptic Laplacian

The 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.