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

:$$, W(x) = \mathrm V(x) W(x) - \mathrm W(x) V(x),

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

:\begin
&A_ (x)~,\\
& _ (x), A_ (x),\\
& A_ (x), A_ (x) A_ (x)]~,\\
&\quad\vdots\quad
\end
\qquad 0 \leq j_, j_, \ldots, j_ \leq n

linear span, 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)

:$\operatorname dx = A_0(x) \operatorname dt + \sum_^n A_i(x) \circ \operatorname dW_i$
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

:$F = \frac1 \sum_^n A_i^2 + A_0.$

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

:$\begin \dfrac (t, x) = F u(t, x), & t > 0, x \in \mathbf^; \\ u(t, \cdot) \to f, & \text t \to 0; \end$

to have a smooth fundamental solution, i.e. a real-valued function ''p'' (0, +∞) × R^2''d'' → R such that ''p''(''t'', ·, ·) is smooth on R^2''d'' for each ''t'' and

:$u(t, x) = \int_ p(t, x, y) f(y) \, \mathrm y$

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

:$A_ = \sum_^ a_ \frac,$

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

:$\dot = \sum_^ u_ A_(x)$

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

See also

* Malliavin calculus
*Lie algebra

References

* (See the introduction)
*

{{DEFAULTSORT:Hormander's Condition
Partial differential equations
Stochastic differential equations