For Au = f with an elliptic differential operator A : H → H′ and stochastic data f , the m-point correlation function Mmu of the random solution u satisfies a deterministic, hypoelliptic equation with the m-fold tensor product operator A of A. Sparse tensor products of hierarchic FE-spaces in H are known to allow for approximations to Mmu which converge at essentially the rate as in the case m ...

For a wide class of continuous-time Markov processes, including all irreducible hypoelliptic diffusions evolving on an open, connected subset of R, the following are shown to be equivalent: (i) The process satisfies (a slightly weaker version of) the classical Donsker-Varadhan conditions; (ii) The transition semigroup of the process can be approximated by a finite-state hidden Markov model, in ...

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on $\mathbb{T}^1 \times \mathbb{S}^3$. the case real-valued coefficients, prove that an operator in is conjugated constant-coefficient satisfying Diophantine condition, such conjugation preserves hypoellipticity. where imaginary part coefficients non...

Partial differential equations. — Gaussian estimates for hypoelliptic operators via optimal control, by UGO BOSCAIN and SERGIO POLIDORO, communicated on 11 May 2007. ABSTRACT. — We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the value function of a suitab...

We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were in...

A semiclassical analysis of a nonlinear eigenvalue problem arising from the study of the failure of analytic hypoellipticity is given. A general family of hypoelliptic, but not analytic hypoelliptic operators is obtained. §

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non selfadjoint operators has a non zero 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.

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. It is well known that L admits a fundamental solution Γ . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique...

Recently, Kohn constructed examples of sums of squares of complex vector fields satisfying Hörmander’s condition that lose derivatives, but are nevertheless hypoelliptic. He also demonstrated optimal L2 regularity. In this paper, we construct parametricies for Kohn’s operators, which lead to the corresponding Lp (1 < p < ∞) and Lipschitz regularity. In fact, our parametrix construction generali...