## Kyoto Journal of Mathematics

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

#### Abstract

Given the standard Gaussian measure $\gamma$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g\cdot\gamma$ absolutely continuous with respect to $\gamma$, we consider the optimal transportation $T(x)=x+\nabla\varphi(x)$ of $g\cdot\gamma$ to $\gamma$. Assume that the function $|\nabla g|^{2}/g$ is $\gamma$-integrable. We prove that the function $\varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g=\operatorname{det}_{2}(\mathrm{I}+D^{2}\varphi)\exp (\mathcal{L}\varphi-\frac{1}{2}|\nabla\varphi|^{2})$. We also establish sufficient conditions for the existence of third-order derivatives of $\varphi$.

#### Article information

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

Dates
First available in Project Euclid: 21 November 2013

https://projecteuclid.org/euclid.kjm/1385042732

Digital Object Identifier
doi:10.1215/21562261-2366078

Mathematical Reviews number (MathSciNet)
MR3160599

Zentralblatt MATH identifier
1286.28011

#### 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

#### References

• [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Wasserstein Spaces of Probability Measures, Lectures Math. ETH Zurich, Birkhäuser, Basel, 2005.
• [2] V. I. Bogachev, Gaussian Measures, Math. Surveys Monogr. 62, Amer. Math. Soc., Providence, 1998.
• [3] V. I. Bogachev, Measure Theory, Vols. I, II, Springer, Berlin, 2007.
• [4] V. I. Bogachev, Differentiable Measures and the Malliavin Calculus, Math. Surveys Monogr. 164, Amer. Math. Soc., Providence, 2010.
• [5] V. I. Bogachev and A. V. Kolesnikov, Nonlinear transformations of convex measures and entropy of the Radon–Nikodym densities (in Russian), Dokl. Acad. Nauk. 397 (2004), 155–159; English translation in Dokl. Math. 70 (2004), 524–528.
• [6] V. I. Bogachev and A. V. Kolesnikov, Integrability of absolutely continuous transformations of measures and applications to optimal mass transport (in Russian), Teor. Verojatn. Primen. 50 (2005), 433–456; English translation in Theory Probab. Appl. 50 (2006), 367–385.
• [7] V. I. Bogachev and A. V. Kolesnikov, Nonlinear transformations of convex measures (in Russian), Teor. Verojatn. Primen. 50 (2005), 27–51; English translation in Theory Probab. Appl. 50 (2006), 34–52.
• [8] V. I. Bogachev and A. V. Kolesnikov, On the Monge–Ampère equation in infinite dimensions, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), 547–572.
• [9] V. I. Bogachev and A. V. Kolesnikov, On the Monge–Ampère equation on Wiener space (in Russian), Dokl. Acad. Nauk. 406 (2006), 7–11; English translation in Dokl. Math. 73 (2006), 1–5.
• [10] V. I. Bogachev, A. V. Kolesnikov, and K. V. Medvedev, Triangular transformations of measures, Sb. Math. 196 (2005), 309–335.
• [11] F. Cavalletti, The Monge problem in Wiener space, Calc. Var. Partial Differential Equations 45 (2012), 101–124.
• [12] D. Feyel and A. S. Üstünel, The notions of convexity and concavity on Wiener space, J. Funct. Anal. 176 (2000), 400–428.
• [13] D. Feyel and A. S. Üstünel, Monge–Kantorovitch measure transportation and Monge–Ampère equation on Wiener space, Probab. Theory Related Fields 128 (2004), 347–385.
• [14] A. V. Kolesnikov, Convexity inequalities and optimal transport of infinite-dimensional measures, J. Math. Pures Appl. (9) 83 (2004), 1373–1404.
• [15] A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, to appear in Theory Probab. Appl., preprint, arXiv:1007.1103v3 [math.PR].
• [16] R. J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), 153–179.
• [17] R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton Univ. Press, Princeton, 1970.
• [18] I. Shigekawa, Stochastic Analysis, Trans. Math. Monogr. 224, Amer. Math. Soc., Providence, 2004.
• [19] A. S. Üstünel and M. Zakai, Transformation of Measure on Wiener Space, Springer Monogr. Math., Springer, Berlin, 2000.
• [20] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, Amer. Math. Soc., Providence, 2003.