Churchill–Bernstein equation: Difference between revisions
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In [[mathematics]] — specifically, in [[measure theory]] — '''Malliavin's absolute continuity lemma''' is a result due to the [[France|French]] [[mathematician]] [[Paul Malliavin]] that plays a foundational rôle in the regularity ([[smooth function|smoothness]]) [[theorem]]s of the [[Malliavin calculus]]. Malliavin's lemma gives a sufficient condition for a [[finite measure|finite]] [[Borel measure]] to be [[absolute continuity|absolutely continuous]] with respect to [[Lebesgue measure]]. | |||
==Statement of the lemma== | |||
Let ''μ'' be a finite Borel measure on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>. Suppose that, for every ''x'' ∈ '''R'''<sup>''n''</sup>, there exists a constant ''C'' = ''C''(''x'') such that | |||
:<math>\left| \int_{\mathbf{R}^{n}} \mathrm{D} \varphi (y) (x) \, \mathrm{d} \mu(y) \right| \leq C(x) \| \varphi \|_{\infty}</math> | |||
for every ''C''<sup>∞</sup> function ''φ'' : '''R'''<sup>''n''</sup> → '''R''' with [[compact support]]. Then ''μ'' is absolutely continuous with respect to ''n''-dimensional Lebesgue measure ''λ''<sup>''n''</sup> on '''R'''<sup>''n''</sup>. In the above, D''φ''(''y'') denotes the [[Fréchet derivative]] of ''φ'' at ''y'' and ||''φ''||<sub>∞</sub> denotes the [[supremum norm]] of ''φ''. | |||
==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 section 1.3) | |||
* {{cite book | |||
| last = Malliavin | |||
| first = Paul | |||
| authorlink = Paul Malliavin | |||
| chapter = Stochastic calculus of variations and hypoelliptic operators | |||
| title = Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) | |||
| pages= 195–263 | |||
| publisher = Wiley | |||
| location = New York | |||
| year = 1978 | |||
}} {{MathSciNet|id=536013}} | |||
[[Category:Lemmas]] | |||
[[Category:Measure theory]] | |||
Revision as of 13:47, 12 January 2014
In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus. Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.
Statement of the lemma
Let μ be a finite Borel measure on n-dimensional Euclidean space Rn. Suppose that, for every x ∈ Rn, there exists a constant C = C(x) such that
for every C∞ function φ : Rn → R with compact support. Then μ is absolutely continuous with respect to n-dimensional Lebesgue measure λn on Rn. In the above, Dφ(y) denotes the Fréchet derivative of φ at y and ||φ||∞ denotes the supremum norm of φ.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet (See section 1.3) - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet