Hörmander's condition

Summary

Hörmander's condition is a property of a family of vector fields. It says that the fields and their iterated Lie brackets span the tangent space at every point. Hörmander's theorem says that under this condition the second-order operator $\sum_j X_j^2 + X_0$ is hypoelliptic, even though it may be far from elliptic. The condition is central in sub-Riemannian geometry and in stochastic analysis, where it gives smooth densities for solutions of stochastic differential equations.