# Hörmander's condition

In mathematics, **Hörmander's condition** is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations. The condition is named after the Swedish mathematician Lars Hörmander.

## Definition

Given two *C*^{1} vector fields *V* and *W* on *d*-dimensional Euclidean space **R**^{d}, let [*V*, *W*] denote their Lie bracket, another vector field defined by

where D*V*(*x*) denotes the Fréchet derivative of *V* at *x* ∈ **R**^{d}, which can be thought of as a matrix that is applied to the vector *W*(*x*), and *vice versa*.

Let *A*_{0}, *A*_{1}, ... *A*_{n} be vector fields on **R**^{d}. They are said to satisfy **Hörmander's condition** if, for every point *x* ∈ **R**^{d}, the vectors

span **R**^{d}. They are said to satisfy the **parabolic Hörmander condition** if the same holds true, but with the index taking only values in 1,...,n.

Now consider the stochastic differential equation

where the vectors fields are assumed to have bounded derivative. Hörmander's theorem asserts that if the SDE above satisfies the parabolic Hörmander condition, then its solutions admit a smooth density with respect to Lebesgue measure.

## Application to the Cauchy problem

With the same notation as above, define a second-order differential operator *F* by

An important problem in the theory of partial differential equations is to determine sufficient conditions on the vector fields *A*_{i} for the Cauchy problem

has a smooth fundamental solution, i.e. a real-valued function *p* (0, +∞) × **R**^{2d}→**R** such that *p*(*t*, ·, ·) is smooth on **R**^{2d} for each *t* and

satisfies the Cauchy problem above. It had been known for some time that a smooth solution exists in the elliptic case, in which

and the matrix *A* = (*a*_{ji}), 1 ≤ *j* ≤ *d*, 1 ≤ *i* ≤ *n* is such that *AA*^{∗} is everywhere an invertible matrix.

The great achievement of Hörmander's 1967 paper was to show that a smooth fundamental solution exists under a considerably weaker assumption: the parabolic version of the condition that now bears his name.

## See also

## References

- Bell, Denis R. (2006).
*The Malliavin calculus*. Mineola, NY: Dover Publications Inc. pp. x+113. ISBN 0-486-44994-7. MR 2250060 (See the introduction) - Hörmander, Lars (1967). "Hypoelliptic second order differential equations".
*Acta Math*.**119**: 147–171. doi:10.1007/BF02392081. ISSN 0001-5962. MR 0222474