Hörmander's condition

Given two C¹ vector fields V and W on d-dimensional Euclidean space R^d, let [V, W] denote their Lie bracket, another vector field defined by

${\displaystyle [V,W](x)=\mathrm {D} V(x)W(x)-\mathrm {D} W(x)V(x),}$

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 A₀, A₁, ... 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

${\displaystyle {\begin{aligned}&A_{j_{0}}(x)~,\\&[A_{j_{0}}(x),A_{j_{1}}(x)]~,\\&[[A_{j_{0}}(x),A_{j_{1}}(x)],A_{j_{2}}(x)]~,\\&\quad \vdots \quad \end{aligned}}\qquad 0\leq j_{0},j_{1},\ldots ,j_{n}\leq n}$

span R^d. They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index ${\displaystyle j_{0}}$ taking only values in 1,...,n.

Consider the stochastic differential equation (SDE)

${\displaystyle \operatorname {d} x=A_{0}(x)\operatorname {d} t+\sum _{i=1}^{n}A_{i}(x)\circ \operatorname {d} W_{i}}$

where the vectors fields ${\displaystyle A_{0},\dotsc ,A_{n}}$ are assumed to have bounded derivative, ${\displaystyle (W_{1},\dotsc ,W_{n})}$ the normalized n-dimensional Brownian motion and ${\displaystyle \circ \operatorname {d} }$ 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.

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

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

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

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

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

${\displaystyle A_{i}=\sum _{j=1}^{d}a_{ji}{\frac {\partial }{\partial x_{j}}},}$

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.

Let M be a smooth manifold and ${\displaystyle A_{0},\dotsc ,A_{n}}$ be smooth vector fields on M. Assuming that these vector fields satisfy Hörmander's condition, then the control system

${\displaystyle {\dot {x}}=\sum _{i=1}^{n}u_{i}A_{i}(x)}$

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

• Malliavin calculus
• Lie algebra

• Bell, Denis R. (2006). The Malliavin calculus. Mineola, NY: Dover Publications Inc. pp. x+113. ISBN 0-486-44994-7. MR2250060 (See the introduction)
• Hörmander, Lars (1967). "Hypoelliptic second order differential equations". Acta Math. 119: 147–171. doi:10.1007/BF02392081. ISSN 0001-5962. MR0222474