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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of
partial Partial may refer to: Mathematics * Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
and stochastic differential equations. The condition is named after the
Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
Lars Hörmander Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal ...
.


Definition

Given two ''C''1 vector fields ''V'' and ''W'' on ''d''-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
R''d'', let 'V'', ''W''denote their Lie bracket, another vector field defined by :
, W The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
(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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
''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, +∞) × R2''d'' → R such that ''p''(''t'', ·, ·) is smooth on R2''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 In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the ...
. See
Orbit (control theory) The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory. Definition Let \dot q=f(q,u) be a \ ^\infty control system, where belongs to a finite-dimensional manifol ...
.


See also

* Malliavin calculus *
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...


References

* (See the introduction) * {{DEFAULTSORT:Hormander's Condition Partial differential equations Stochastic differential equations