We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and usi...

We give a sufficient condition on a Lévy measure μ which ensures that the generator L of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., sing suppu ⊆ sing suppLu for all admissible u. In particular, we assume that μ|Rd\{0} ∈ C ∞(Rd \ 0). We also show that this condition is necessary provided that suppμ is compact.

(1) ∂̄b = ∂x + i(∂y − φ(x)∂t) where φ ∈ C∞(R) is real-valued. Such CR structures may be realized as the boundaries of tube domains {z : Im z2 > φ(Re z1)} in C. The Levi form may be identified with the function φ′′(x). We always assume that φ is convex, so that the structure is pseudoconvex. By ∂̄∗ b we mean the adjoint of ∂̄b with respect to L (R, dx dy dt); thus ∂̄∗ b = −∂x + i(∂y − φ(x)∂t). The p...

In this paper we consider the regularity of solutions of semilinear differential equations principal parts of which consist of linear polynomial operators constructed from real vector fields. Based on the study of fine properties of the principal linear parts we then obtain the regularity of solutions of the nonlinear equations. Some results obtained in this article are also new in the frame of...