Kyoto Journal of Mathematics

Sobolev regularity for the Monge–Ampère equation in the Wiener space

Vladimir I. Bogachev and Alexander V. Kolesnikov

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Abstract

Given the standard Gaussian measure γ on the countable product of lines R and a probability measure gγ absolutely continuous with respect to γ, we consider the optimal transportation T(x)=x+φ(x) of gγ to γ. Assume that the function |g|2/g is γ-integrable. We prove that the function φ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula g=det2(I+D2φ)exp(Lφ12|φ|2). We also establish sufficient conditions for the existence of third-order derivatives of φ.

Article information

Source
Kyoto J. Math., Volume 53, Number 4 (2013), 713-738.

Dates
First available in Project Euclid: 21 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1385042732

Digital Object Identifier
doi:10.1215/21562261-2366078

Mathematical Reviews number (MathSciNet)
MR3160599

Zentralblatt MATH identifier
1286.28011

Subjects
Primary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]
Secondary: 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12] 58E99: None of the above, but in this section 60H07: Stochastic calculus of variations and the Malliavin calculus

Citation

Bogachev, Vladimir I.; Kolesnikov, Alexander V. Sobolev regularity for the Monge–Ampère equation in the Wiener space. Kyoto J. Math. 53 (2013), no. 4, 713--738. doi:10.1215/21562261-2366078. https://projecteuclid.org/euclid.kjm/1385042732


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References

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