# 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
    $[V,W](x)=\mathrm {D} V(x)W(x)-\mathrm {D} W(x)V(x),$
where DV(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
    ${\begin{aligned}&A_{j_{0}}(x)~,\\&[A_{j_{0}}(x),A_{j_{1}}(x)]~,\\&[[A_{j_{0}}(x),A_{j_{1}}(x)],A_{j_{2}}(x)]~,\\&\quad \vdots \quad \end{aligned}}\qquad 0\leq j_{0},j_{1},\ldots ,j_{n}\leq n$
span R$^d$. They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index $j_{0}$ taking only values in 1,...,n.

## Application to stochastic differential equations

Consider the stochastic differential equation (SDE)
    $\operatorname {d} x=A_{0}(x)\operatorname {d} t+\sum _{i=1}^{n}A_{i}(x)\circ \operatorname {d} W_{i}$
where the vectors fields $A_{0},\dotsc ,A_{n}$ are assumed to have bounded derivative, $(W_{1},\dotsc ,W_{n})$ the normalized n-dimensional Brownian motion and $\circ \operatorname {d}$ stands for the Stratonovich integral interpretation of the SDE. 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
    $F={\frac {1}{2}}\sum _{i=1}^{n}A_{i}^{2}+A_{0}.$
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
    ${\begin{cases}{\dfrac {\partial u}{\partial t}}(t,x)=Fu(t,x),&t>0,x\in \mathbf {R} ^{d};\\u(t,\cdot )\to f,&{\text{as }}t\to 0;\end{cases}}$
to have 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
    $u(t,x)=\int _{\mathbf {R} ^{d}}p(t,x,y)f(y)\,\mathrm {d} y$
satisfies the Cauchy problem above. It had been known for some time that a smooth solution exists in the elliptic case, in which
    $A_{i}=\sum _{j=1}^{d}a_{ji}{\frac {\partial }{\partial x_{j}}},$
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.

## Application to control systems

Let M be a smooth manifold and $A_{0},\dotsc ,A_{n}$ be smooth vector fields on M. Assuming that these vector fields satisfy Hörmander's condition, then the control system
    ${\dot {x}}=\sum _{i=0}^{n}u_{i}A_{i}(x)$
is locally controllable in any time at every point of M. This is known as the Chow–Rashevskii theorem. See Orbit (control theory).