en>MvH: /* Applications */ added a reference

2014-02-06T13:23:12Z

Applications: added a reference

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	~~In [[mathematics]], '''Hörmander's condition'''~~ is a ~~property of [[vector field]]s that, if satisfied, has many useful consequences~~ in ~~the theory of [[partial differential equation\|partial]] and [[stochastic differential equation]]s~~. ~~The condition~~ is ~~named after the [[Sweden\|Swedish]] [[mathematician]] [[Lars Hörmander]]~~.		Wilber Berryhill is what his spouse enjoys to contact him and he completely enjoys this name. He functions as a bookkeeper. Her family members lives in Alaska but her spouse wants them to transfer. To play lacross is something I really enjoy doing.<br><br>Here is my web blog ... psychic love readings; [http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 www.january-yjm.com],

	~~==Definition==~~
	~~Given two [[smooth function\|''C''~~<~~sup~~>1<~~/sup~~> vector fields]] ''V'' and ''W'' on ''d''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''d''</sup>, let [''V'', ''W''] denote their [[Lie bracket of vector fields\|Lie bracket]], another vector field defined by

	~~:<math>[V, W] (x) = \mathrm{D} V(x) W(x) - \mathrm{D} W(x) V(x),</math>~~

	~~where D''V''(''x'') denotes the [[Fréchet derivative]] of ''V'' at ''x'' ∈ '''R'''<sup>''d''</sup>, which can be thought of as a [[matrix (mathematics)\|matrix]] that~~ is ~~applied to the vector ''W''(''x''), and ''vice versa''~~.

	~~Let ''A''<sub>0</sub>, ''A''<sub>1</sub>,~~ ..~~. ''A''<sub>''n''</sub> be vector fields on '''R'''<sup>''d''</sup>. They are said to satisfy '''Hörmander's condition''' if, for every point ''x''&nbsp~~;~~∈ '''R'''<sup>''d''</sup>, the vectors~~

	:~~<math>\begin{align}~~
	~~&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{align}~~
	~~\qquad 0 \leq j_{0}, j_{1}, \ldots, j_{n} \leq n~~
	</~~math>~~

	~~[[linear span\|span]] '''R'''<sup>''d''<~~/~~sup>~~. ~~They are said to satisfy the '''parabolic Hörmander condition''' if the same holds true, but with the index <math>j_0</math> taking only values in 1,...,n.~~

	~~==Application to the Cauchy problem==~~
	~~With the same notation as above, define a second~~-~~order [[differential operator]] ''F'' by~~

	~~:<math>F = \frac1{2} \sum_{i = 1}^{n} A_{i}^{2} + A_{0}~~.</~~math>~~

	~~An important problem in the theory of partial differential equations is to determine sufficient conditions on the vector fields ''A''<sub>''i''<~~/~~sub> for the Cauchy problem~~

	~~:<math>\begin{cases} \dfrac{\partial u}{\partial t} (t, x) = F u(t, x), & t > 0, x \in \mathbf{R}^{d}; \\ u(t, \cdot) \to f, & \mbox{ as } t \to 0; \end{cases}</math>~~

	~~has a smooth [[fundamental solution]], i~~.e. a real-valued function ''p'' (0, +∞) × '''R'''<sup>2''d''</sup>→'''R''' such that ''p''(''t'', ·, ·) is smooth on '''R'''<sup>2''d''</sup> for each ''t'' and

	~~:<math>u(t, x)~~ = ~~\int_{\mathbf{R}^{d}} p(t, x, y) f(y) \, \mathrm{d} y</math>~~

	~~satisfies the Cauchy problem above. It had been known for some time that a smooth solution exists in the [[elliptic operator\|elliptic]] case, in which~~

	~~:<math>A_{i} = \sum_{j = 1}^{d} a_{ji} \frac{\partial}{\partial x_{j}},</math>~~

	~~and the matrix ''A''~~&~~nbsp;~~=~~ (''a''<sub>''ji''</sub>), 1 ≤ ''j'' ≤ ''d'', 1 ≤ ''i'' ≤ ''n'' is such that ''AA''<sup>∗</sup> 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.

	~~==See also==~~
	*[[Malliavin calculus]]
	*[[Lie Algebra]]

	~~==References==~~
	* {{cite book
	~~\| last = Bell~~
	~~\| first = Denis R.~~
	~~\| title = The Malliavin calculus~~
	~~\| publisher = Dover Publications Inc.~~
	~~\| location = Mineola~~, NY
	~~\| year = 2006~~
	~~\| pages = 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}}~~

	~~{{DEFAULTSORT:Hormander's Condition}}~~
	~~[[Category:Partial differential equations]]~~
	~~[[Category:Stochastic differential equations]]~~

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

2014-02-04T06:01:14Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

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en>MvH: /* Applications */ added a reference

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

174.16.162.24: 404 ("Not Found") Error on Cipra article