Top Qs
Timeline
Chat
Perspective

Malliavin's absolute continuity lemma

Result in measure theory From Wikipedia, the free encyclopedia

Remove ads

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

Summarize
Perspective

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.

Remove ads

References

  • Bell, Denis R. (2006). The Malliavin calculus. Mineola, NY: Dover Publications Inc. pp. x+113. ISBN 0-486-44994-7. MR2250060 (See section 1.3)
  • Malliavin, Paul (1978). "Stochastic calculus of variations and hypoelliptic operators". Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). New York: Wiley. pp. 195–263. MR536013
Remove ads
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads