Hypoelliptic Equations Research Articles

The method of stochastic characteristics for linear second-order hypoelliptic equations

We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green’s functions and to obtain a generalization of Bony’s Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.

A Probabilistic Point of View for the Kolmogorov Hypoelliptic Equations

In this work, we propose a method for solving Kolmogorov hypoelliptic equations based on Fourier transform and Feynman–Kac formula. We first explain how the Feynman–Kac formula can be used to compute the fundamental solution to parabolic equations with linear or quadratic potential. Then applying these results after a Fourier transform we deduce the computation of the solution to a first class of Kolmogorov hypoelliptic equations. Then we solve partial differential equations obtained via Feynman–Kac formula from the Ornstein–Uhlenbeck generator. Also, a new small time approximation of the solution to a certain class of Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation.

Quantitative De Giorgi methods in kinetic theory

We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{o}lder continuity with quantitative estimates. The paper is self-contained.

Tunneling estimates and approximate controllability for hypoelliptic equations

This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator L \mathcal {L} on a compact manifold M \mathcal {M} assuming: ( i ) (i) the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of L \mathcal {L} , and ( i i ) (ii) the analyticity of M \mathcal {M} and the coefficients of L \mathcal {L} . The first result is the tunneling estimate ‖ φ ‖ L 2 ( ω ) ≥ C e − c λ k 2 \|\varphi \|_{L^2(\omega )} \geq Ce^{- c\lambda ^{\frac {k}{2}}} for normalized eigenfunctions φ \varphi of L \mathcal {L} from a nonempty open set ω ⊂ M \omega \subset \mathcal {M} , where k k is the hypoellipticity index of L \mathcal {L} and λ \lambda the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation ( ∂ t 2 + L ) u = 0 (\partial _t^2+\mathcal {L})u=0 : for T > 2 sup x ∈ M ( d i s t ( x , ω ) ) T>2 \sup _{x \in \mathcal {M}}(dist(x,\omega )) (here, d i s t dist is the sub-Riemannian distance), the observation of the solution on ( 0 , T ) × ω (0,T)\times \omega determines the data. The constant involved in the estimate is C e c Λ k Ce^{c\Lambda ^k} where Λ \Lambda is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation ( ∂ t + L ) v = 1 ω f (\partial _t+\mathcal {L})v=\mathbb {1}_\omega f in any time, with appropriate (exponential) cost, depending on k k . In case k = 2 k=2 (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary ∂ M \partial \mathcal {M} can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).

Some global Sobolev inequalities related to Kolmogorov-type operators

In this note we review a recent result in [17] in collaboration with N. Garofalo, where we establish global versions of Hardy-Littlewood-Sobolev inequalities attached to hypoelliptic equations of Kolmogorov type. The relevant Sobolev spaces are defined through the fractional powers of the operator under consideration. We outline the main steps of the semigroup approach we adopt.

Quantitative unique continuation for hyperbolic and hypoelliptic equations

We review recent results of the authors concerning quantitative unique continuation estimates for operators with coefficients that are analytic in some (or all the) variables. We describe several applications for wave-like equations, but also equations based on hypoelliptic operators. These proceedings are a survey of the general results in [LL19] together with applications to wave equations [LL16] and to hypoelliptic equations [LL17, LL20b].

On the properties of the symbols of one class of hypoelliptic equations

On the properties of the symbols of one class of hypoelliptic equations

Application of Nilpotent Approximation and the Orbit Method to the Search of the Diagonal Asymptotics of Sub-Riemannian Heat Kernels

We propose a general scheme for the search of a fundamental solution to the hypoelliptic diffusion equation in a “sufficiently good” sub-Riemannian manifold and the small-time asymptotics for the solution, which includes the generalized Fourier transform and the orbit method closely related to it, as well as an application of the perturbative method to the nilpotent approximation, and Trotter’s formula.

On the Properties of the Symbols of One Class of Hypoelliptic Equations

We consider regular hypoelliptic operators and study some properties of completely regular polyhedra. Basing on the obtained properties, we find an upper bound for the functional dimension of the solution spaces of hypoelliptic equations.

Functional inequalities for a class of nonlocal hypoelliptic equations of Hörmander type

We consider a class of second-order partial differential operators A of Hörmander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the ’30s. We analyse some properties of the nonlocal operators driven by the fractional powers of A, and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincaré inequalities adapted to the underlying geometry.

Correct Solvability of the Dirichlet Problem in the Half-space for Regular Hypoelliptic Equations

In this paper we study the Dirichlet problem in the half-space for regular hypoelliptic equations. Applying a special integral representation, we construct approximate solutions for this problem and thereby prove correct solvability of the problem.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Green’s Identities, Comparison Principle and Uniqueness of Positive Solutions for Nonlinear p-sub-Laplacian Equations on Stratified Lie Groups

We propose analogues of Green’s and Picone’s identities for the p-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Diaz-Saa inequality. Using these we derive a comparison principle and uniqueness of positive solutions to nonlinear hypoelliptic equations on general stratified Lie groups extending to this setting previously known results on Euclidean and Heisenberg groups.

ON SMOOTH SOLUTIONS OF A CLASS OF ALMOST HYPOELLIPTIC EQUATIONS OF CONSTANT STRENGTH

ON SMOOTH SOLUTIONS OF A CLASS OF ALMOST HYPOELLIPTIC EQUATIONS OF CONSTANT STRENGTH

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

On Solvability of Regular Hypoelliptic Equations in ℝn

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.

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x1,... x n , D1,..., D n ) = ∑αγα(x)Dα with coefficients γα(x) defined in E n is called formally almost hypoelliptic in E n if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.

A special class of nonlinear hypoelliptic equations on spheres

We study nonlinear second-order equations of the form \(-\Delta u + X^2 u + i\lambda X u + \sigma u = K|u|^{p - 1}u\) on \(S^n\) with the usual round metric, where X is a Killing field. The case of principal interest is when X has length \({\le }1\), which leads to hypoelliptic operators with loss of one derivative. After establishing existence of solutions via variational methods, we carry out the subelliptic analysis on \(S^n\) utilizing their homogeneous coset space properties. Writing \(L_\alpha = \Delta - X^2 - i\alpha X\), we establish the optimal range of p such that the embedding \(\mathcal {D}((-L_\alpha )^{1/2}) \hookrightarrow L^p\) is compact, which gives sharp versions of results in Taylor (Houst J Math 42(1):143–165, 2016) for all dimensions. Such subelliptic phenomena have no parallel in the setting of flat spaces.

Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

We extend the De Giorgi–Nash–Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the Holder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations of type II , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law γ ∈ [−d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the Holder regularity of these solutions.

Existence theorem and global solution for semilinear edge-degenerate hypoelliptic equations

In this article, we use the edge-type of Sobolev inequality,Hardy inequlity and Poincare inequality to prove the existence theorem for a class of semilinear degenerate hypoelliptic equation on manifolds with conical singularities. In this paper we shall find the existence theorem for the problem 1.1 in cone Sobolev space \({\mathcal {H}}^{1,\frac{N}{2}}_{2,0}({\mathbb {E}}).\) Finally, we obtain existence result of global solutions with exponential decay and show the blow-up in finite time of solutions.