Let {X1, . . . ,Xp} be complex-valued vector fields in Rn and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields). Our object is to study the operator E = ∑ X∗ i Xi, where X ∗ i is the L2 adjoint of Xi. A result of Hörmander is that when the Xi are real then E is hypoelliptic and furthemore it is subelliptic (the restriction of a destribution u t...

We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological K-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a K-cocycle has already been constructed by Boutet de Monvel for Toeplitz operators, a...

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to product of fractals. While our resul...

We consider Ornstein-Uhlenbeck processes (OU-processes) related to hypoelliptic diffusion on finite-dimensional Lie groups: let L be a hypoelliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-type processes by adding horizontal gradient drifts of functions U . In the natural case U(x) = − log p(1, x), where p(1, x) is the...

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: L = ∑m i=1X 2 i + ∆ in Rn, where ∆ is the Laplace operator, m < n, and the limit operator L = ∑m i=1X 2 i is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter , of solution of the approximated equation L u = f , using a modification of the lift...

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-sca...

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 t...

We consider an operator P which is a sum of squares of vector fields with analytic coefficients. The operator has a nonsymplectic characteristic manifold, but the rank of the symplectic form σ is not constant on CharP . Moreover the Hamilton foliation of the non symplectic stratum of the Poisson-Treves stratification for P consists of closed curves in a ring-shaped open set around the origin. W...

For a second-order particle system in Rd subject to locally-in-space pairwise annihilation, we prove scaling limit for its empirical measure on position and velocity towards degenerate elliptic partial differential equation. Crucial ingredients are Green’s function estimates the associated hypoelliptic operator an Itô-Tanaka trick.

A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is Fredholm operator. Its symbol determines an element K-theory of noncommutative algebra symbols. We construct periodic cyclic cocycle which, when paired with Connes-Chern character principal symbol, calculates index. Our index formula local, i.e. given as local expression terms and connection TM its curvature. pr...