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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]].
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| ==Definition==
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| 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
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| :<math>[V, W] (x) = \mathrm{D} V(x) W(x) - \mathrm{D} W(x) V(x),</math>
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| 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''.
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| 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'' ∈ '''R'''<sup>''d''</sup>, the vectors
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| :<math>\begin{align} | |
| &A_{j_0} (x)~,\\
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| &[A_{j_{0}} (x), A_{j_{1}} (x)]~,\\
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| &[[A_{j_{0}} (x), A_{j_{1}} (x)], A_{j_{2}} (x)]~,\\
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| &\quad\vdots\quad
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| \end{align}
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| \qquad 0 \leq j_{0}, j_{1}, \ldots, j_{n} \leq n
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| </math>
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| [[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.
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| ==Application to the Cauchy problem==
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| With the same notation as above, define a second-order [[differential operator]] ''F'' by
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| :<math>F = \frac1{2} \sum_{i = 1}^{n} A_{i}^{2} + A_{0}.</math>
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| 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
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| :<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>
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| 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
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| :<math>u(t, x) = \int_{\mathbf{R}^{d}} p(t, x, y) f(y) \, \mathrm{d} y</math>
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| 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
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| :<math>A_{i} = \sum_{j = 1}^{d} a_{ji} \frac{\partial}{\partial x_{j}},</math>
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| and the matrix ''A'' = (''a''<sub>''ji''</sub>), 1 ≤ ''j'' ≤ ''d'', 1 ≤ ''i'' ≤ ''n'' is such that ''AA''<sup>∗</sup> is everywhere an [[invertible matrix]].
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| 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.
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| ==See also==
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| *[[Malliavin calculus]]
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| *[[Lie Algebra]]
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| ==References==
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| * {{cite book
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| | last = Bell
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| | first = Denis R.
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| | title = The Malliavin calculus
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| | publisher = Dover Publications Inc.
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| | location = Mineola, NY
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| | year = 2006
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| | pages = x+113
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| | isbn = 0-486-44994-7
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| }} {{MathSciNet|id=2250060}} (See the introduction)
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| * {{cite journal
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| | last = Hörmander
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| | first = Lars
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| | authorlink = Lars Hörmander
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| | title = Hypoelliptic second order differential equations
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| | journal = Acta Math.
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| | volume = 119
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| | year = 1967
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| | pages = 147–171
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| | issn = 0001-5962
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| | doi = 10.1007/BF02392081
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| }} {{MathSciNet|id=0222474}}
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| {{DEFAULTSORT:Hormander's Condition}}
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| [[Category:Partial differential equations]]
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| [[Category:Stochastic differential equations]]
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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.
Here is my web blog ... psychic love readings; www.january-yjm.com,