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 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

:$A\_\{i\}\; (x),$:$[A\_\{j\_\{0\; (x),\; A\_\{j\_\{1\; (x)]\; ,$:$[A\_\{j\_\{0\; (x),\; A\_\{j\_\{1\; (x)]\; ,\; A\_\{j\_\{2\; (x)]\; ,$::$vdots$:$1\; leq\; i\; leq\; n,\; 0\; leq\; j\_\{0\},\; j\_\{1\},\; ldots,\; j\_\{n\}\; leq\; n,$

span R^"d".

Application to the Cauchy problem

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

:$F\; =\; frac1\{2\}\; sum\_\{i\; =\; 1\}^\{n\}\; A\_\{i\}^\{2\}\; +\; A\_\{0\}.$

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

:

has a smooth fundamental solution, i.e. a real-valued function "p" (0, +∞) × R^2"d" such that "p"("t", ·, ·) is smooth on R^2"d" 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 condition that now bears his name.

References

* cite book
last = Bell
first = Denis R.
title = The Malliavin calculus
publisher = Dover Publications Inc.
location = Mineola, NY
year = 2006
pages = pp. x+113
isbn = 0-486-44994-7 MathSciNet|id=2250060 (See the introduction)
* cite journal
last = Hörmander
first = Lars
authorlink = Lars Hörmander
title = Hypoelliptic second order differential equations
journal = Acta Math.
volume = 119
year = 1967
pages = 147–171
issn = 0001-5962
doi = 10.1007/BF02392081 MathSciNet|id=0222474