Hörmander's condition

Application to stochastic differential equations

Consider the stochastic differential equation (SDE)

${\displaystyle \mathrm{d}x=A_0(x)\mathrm{d}t+\sum _{i=1}^n{A}_i(x)\circ \mathrm{d}{W}_i}$

where the vectors fields ${\displaystyle A_0,\dots ,A_n}$ are assumed to have bounded derivative, ${\displaystyle (W_1,\dots ,W_n)}$ the normalized n-dimensional Brownian motion and ${\displaystyle \circ \mathrm{d}}$ 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

${\displaystyle F=\frac{1}{2}\sum _{i=1}^n{A}_i^2+A_0.}$

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

${\displaystyle \{\begin{array}{ll}{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}(t,x)=Fu(t,x),& t>0,x\in {\mathbf{R}}^d;\\ u(t,\cdot )\to f,& \text{as }t\to 0;\end{array}}$

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

${\displaystyle u(t,x)={\int }_{{\mathbf{R}}^d}p(t,x,y)f(y)\mathrm{d}y}$

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

${\displaystyle A_i=\sum _{j=1}^d{a}_{ji}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j},}$

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 ${\displaystyle A_0,\dots ,A_n}$ be smooth vector fields on M. Assuming that these vector fields satisfy Hörmander's condition, then the control system

${\displaystyle \dot{x}=\sum _{i=0}^n{u}_i{A}_i(x)}$

is locally controllable in any time at every point of M. This is known as the Chow–Rashevskii theorem. See Orbit (control theory).