In this paper we study properties of symbols such that these belong to class of symbols sitting insideSm ρ,φ that we shall introduce as the following. So for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of M−hypoelliptic pseudodifferential operators for which define base on this class of symbols. Also we consider maxi...

Certain second-order partial differential operators, which are expressed as sums of squares of real-analytic vector fields in R3 and which are well known to be C hypoelliptic, fail to be analytic hypoelliptic.

This work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. ...

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of asymptotic average solutions making pointwise solvable every Poisson equation ${\mathcal {L}} u(x)=-f(x)$ with continuous data f, where {L}}$ is hypoelliptic linear partial differential operator positive semidefinite characteristic form.

We derive the hypoelliptic estimates for a kinetic equation of the form ∂tf + k · ∇xf = (−∆d)h, for (t, x, k) ∈ R× R × S, where d ≥ 1, β > 0, Sd is the unit sphere in Rd+1 and ∆d is the Laplace-Beltrami operator on Sd. Such equations arise in the modeling of high frequency waves in random media with long-range correlations. Assuming some (fractional) Sobolev regularity in the momentum variable ...

The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the FreidlinWentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular di...

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in N dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last N−1 variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coeffi...

The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular d...

In [EE1] and [EE2] we presented the solution to the index problem for a natural class of hypoelliptic differential operators on compact contact manifolds. The methods developed to deal with that problem have wider applicability to the index theory of hypoelliptic Fredholm operators. As an example of the power of the proof techniques we present here a new proof of a little known index theorem of...