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This article is cited in 20 scientific papers (total in 20 papers)
Triangular transformations of measures
V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev
M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
A new identity for the entropy of a non-linear image of a measure on $\mathbb R^n$ is obtained, which yields the well-known Talagrand's inequality. Triangular mappings on $\mathbb R^n$ and $\mathbb R^\infty$ are studied, that is, mappings $T$ such that the $i$th coordinate function $T_i$ depends only on the variables $x_1,…,x_i$. With the help of such mappings the well-known open problem on the representability of each probability measure that is absolutely continuous with respect to a Gaussian measure $\gamma$ on an infinite dimensional space as the image of $\gamma$ under a map of the form $T(x)=x+F(x)$ where $F$ takes values in the Cameron–Martin space of the measure $\gamma$ is solved in the affirmative. As an application, a generalized logarithmic Sobolev inequality is also proved.
DOI:
https://doi.org/10.4213/sm1271
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English version:
Sbornik: Mathematics, 2005, 196:3, 309–335
Bibliographic databases:
    
UDC:
519.2
MSC: 28C20, 46G12, 60B11
Received: 27.05.2004
Citation:
V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev, “Triangular transformations of measures”, Mat. Sb., 196:3 (2005), 3–30; Sb. Math., 196:3 (2005), 309–335
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D. E. Aleksandrova, “Convergence of triangular transformations of measures”, Theory Probab. Appl., 50:1 (2006), 113–118
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Kirill V. Medvedev, “Certain properties of triangular
transformations of measures”, Theory Stoch. Process., 14(30):1 (2008), 95–99
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Bogachev V.I., Kolesnikov A.V., “Mass transport generated by a flow of Gauss maps”, J. Funct. Anal., 256:3 (2009), 940–957
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Fang Shizan, Shao Jinghai, Sturm Karl-Theodor, “Wasserstein space over the Wiener space”, Probab. Theory Relat. Fields, 146:3-4 (2010), 535–565
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Zhdanov R.I., “Continuity and differentiability of triangular mappings”, Dokl. Math., 82:2 (2010), 741–745
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Sari Lasanen, “Non-Gaussian statistical inverse problems. Part I: Posterior distributions”, IPI, 6:2 (2012), 215
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A. V. Kolesnikov, “Sobolev regularity of transportation of probability measures and transportation inequalities”, Theory Probab. Appl., 57:2 (2013), 243–264
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V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890
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Mikami T., “A Characterization of the Knothe-Rosenblatt Processes by a Convergence Result”, SIAM J. Control Optim., 50:4 (2012), 1903–1920
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V. I. Bogachev, A. V. Kolesnikov, “Sobolev Regularity for the Monge-Ampere Equation in the Wiener Space”, Kyoto J. Math., 53:4 (2013), 713–738
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A. V. Kolesnikov, “Weak regularity of Gauss mass transport”, Bull. Sci. Math., 138:2 (2014), 165–198
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A. V. Kolesnikov, M. Röckner, “On continuity equations in infinite dimensions with non-Gaussian reference measure”, J. Funct. Anal., 266:7 (2014), 4490–4537
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D. B. Bukin, “On the Kantorovich Problem for Nonlinear Images of the Wiener Measure”, Math. Notes, 100:5 (2016), 660–665
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Backhoff J., Beiglbock M., Lin Y., Zalashko A., “Causal Transport in Discrete Time and Applications”, SIAM J. Optim., 27:4 (2017), 2528–2562
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V. I. Bogachev, A. N. Kalinin, S. N. Popova, “On the equality of values in the Monge and Kantorovich problems”, J. Math. Sci. (N. Y.), 238:4 (2019), 377–389
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Morrison R.E., Baptista R., Marzouk Y., Advances in Neural Information Processing Systems 30 (Nips 2017), Advances in Neural Information Processing Systems, 30, eds. Guyon I., Luxburg U., Bengio S., Wallach H., Fergus R., Vishwanathan S., Garnett R., Neural Information Processing Systems (Nips), 2017
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Spantini A., Bigoni D., Marzouk Y., “Inference Via Low-Dimensional Couplings”, J. Mach. Learn. Res., 19 (2018), 66
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Dmitry V. Bakin, Elena P. Krugova, “Transportation costs for optimal and triangular transformations of Gaussian measures”, Theory Stoch. Process., 23(39):2 (2018), 21–32
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