Ormander Condition Research Articles

A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

In this article, we review our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. This hallmark of chaos has long been observed in these models, however, no mathematical proof had been made for either deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an $L^1$-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies H\"ormander's condition. We review the recent contributions of the first and third author on the verification of this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio. Finally, we briefly contrast this work with our earlier work on Lagrangian chaos in the stochastic Navier-Stokes equations. We end the review with a discussion of some open problems.

Singular stochastic integral operators

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains.

Sparse Domination Theorem for Multilinear Singular Integral Operators with Lr-Hörmander Condition

In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$-H\"ormander condition, then $T$ can be dominated by multilinear sparse operators.

Dynamics of Geodesic Flows with Random Forcing on Lie Groups with Left-Invariant Metrics

We consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups and study the H\"ormander condition and some properties of the solutions of the corresponding Fokker-Planck equations.

Wave equations on graded groups and hypoelliptic Gevrey spaces

The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are H\older-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obtained by Garetto and Ruzhansky in the case of a compact Lie group. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In this investigation, a structure reminiscent of Gevrey regularity appears, inspiring deeper investigation of certain classes of functions and a comparison with Gevrey classes. In the latter part of this thesis we discuss such Gevrey spaces associated to the sums of squares of vector fields satisfying the H\ormander condition on manifolds. This provides a deeper understanding of the Gevrey hypoellipticity of sub-Laplacians. It is known that if $\mathcal L$ is a Laplacian on a closed manifold $M$ then the standard Gevrey space $\gamma^s$ on $M$ defined in local coordinates can be characterised by the condition that $\|e^{D\mathcal L^{\frac{1}{2s}}}\phi\|_{L^2(M)} 0$. The aim in this part is to discuss the conjecture that a similar characterisation holds true if $\mathcal L$ is H\ormander's sum of squares of vector fields, with a sub-Laplacian version of the Gevrey spaces involving these vector fields only. We prove this conjecture in one direction, while in the other we show it holds for sub-Laplacians on $SU(2)$ and on the Heisenberg group $\mathbb{H}_n$.

Densities for SDEs driven by degenerate $\alpha$-stable processes

In this work, by using the Malliavin calculus, under H\"ormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker-Planck (nonlocal) operator: \[\mathscr{L}^{(\alpha)}_b:=\Delta^{\alpha/2}_{\mathrm{v}}+\mathrm {v}\cdot \nabla_x+b(x,\mathrm{v})\cdot \nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^d,\] where $\alpha\in(0,2)$ and $b:\mathbb{R}^d\times\mathbb{R}^d\to \mathbb{R}^d$ is smooth and has bounded derivatives of all orders.

Hölder equivalence of the value function for control-affine systems

We prove the continuity and the H\"older equivalence w.r.t.\ an Euclidean distance of the value function associated with the $L^1$ cost of the control-affine system $\dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q)$, satisfying the strong H\"ormander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the $L^1$ cost of time-dependent systems of the form $\dot q = \sum_{j=1}^m u_j f_j^t(q)$.

Regularity of Wiener functionals under a Hörmander type condition of order one

We study the local existence and regularity of the density of the law of a functional on the Wiener space which satisfies a criterion that generalizes the H\"ormander condition of order one (that is, involving the first order Lie brackets) for diffusion processes.

Lpand Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are H\"older continuous with respect to the distance induced by the vector fields, then local Schauder estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space VMO with respect to the distance induced by the vector fields, then local L^{p} estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the presence of the drift term X_{0}, so that our class of operators covers, for instance, Kolmogorov-Fokker-Planck operators.

Hypoellipticity and Non Hypoellipticity for Sums of Squares of Complex Vector Fields

In this talk we give a report on a paper where we consider a model sum of squares of planar complex vector fields, being close to Kohn's operator but with a point singularity. The characteristic variety of the operator is the same symplectic real analytic manifold as Kohn's. We show that this operator is hypoelliptic and Gevrey hypoelliptic provided certain conditions are satisfied. We show that in the Gevrey spaces below a certain index the operator is not hypoelliptic. Moreover there are cases in which the operator is not even hypoelliptic. This fact leads to some general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex H\ormander condition is satisfied.

A Green function and regularity results for an ultraparabolic equation with a singular potential

We prove a Harnack inequality for the positive solutions of a Schr\"odinger type equation $$ L_0\,u\,+\,V\,u\,=\,0, $$ where $L_0$ is an operator satisfying the H\"ormander's condition and $V$ belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem.