Malliavin's absolute continuity lemma

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

$\left| \int_{\mathbf{R}^{n}} \mathrm{D} \varphi (y) (x) \, \mathrm{d} \mu(y) \right| \leq C(x) \| \varphi \|_{\infty}$

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

• 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