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Mat. Sb., 2005, Volume 196, Number 3, Pages 3–30 (Mi msb1271)  

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.


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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

Citation in format AMSBIB
\by V.~I.~Bogachev, A.~V.~Kolesnikov, K.~V.~Medvedev
\paper Triangular transformations of measures
\jour Mat. Sb.
\yr 2005
\vol 196
\issue 3
\pages 3--30
\jour Sb. Math.
\yr 2005
\vol 196
\issue 3
\pages 309--335

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    This publication is cited in the following articles:
    1. V. I. Bogachev, A. V. Kolesnikov, “Integrability of absolutely continuous measure transformations and applications to optimal transportation”, Theory Probab. Appl., 50:3 (2006), 367–385  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. V. I. Bogachev, A. V. Kolesnikov, “Nonlinear transformations of convex measures”, Theory Probab. Appl., 50:1 (2006), 34–52  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. D. E. Aleksandrova, “Convergence of triangular transformations of measures”, Theory Probab. Appl., 50:1 (2006), 113–118  mathnet  crossref  crossref  mathscinet  isi  elib
    4. Kirill V. Medvedev, “Certain properties of triangular transformations of measures”, Theory Stoch. Process., 14(30):1 (2008), 95–99  mathnet  mathscinet  zmath
    5. Bogachev V.I., Kolesnikov A.V., “Mass transport generated by a flow of Gauss maps”, J. Funct. Anal., 256:3 (2009), 940–957  crossref  mathscinet  zmath  isi  elib
    6. Fang Shizan, Shao Jinghai, Sturm Karl-Theodor, “Wasserstein space over the Wiener space”, Probab. Theory Relat. Fields, 146:3-4 (2010), 535–565  crossref  mathscinet  zmath  isi
    7. Zhdanov R.I., “Continuity and differentiability of triangular mappings”, Dokl. Math., 82:2 (2010), 741–745  crossref  mathscinet  zmath  isi  elib  elib
    8. Sari Lasanen, “Non-Gaussian statistical inverse problems. Part I: Posterior distributions”, IPI, 6:2 (2012), 215  crossref  mathscinet  zmath  isi
    9. A. V. Kolesnikov, “Sobolev regularity of transportation of probability measures and transportation inequalities”, Theory Probab. Appl., 57:2 (2013), 243–264  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. Mikami T., “A Characterization of the Knothe-Rosenblatt Processes by a Convergence Result”, SIAM J. Control Optim., 50:4 (2012), 1903–1920  crossref  mathscinet  zmath  isi  elib
    12. 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  crossref  mathscinet  zmath  isi  elib
    13. A. V. Kolesnikov, “Weak regularity of Gauss mass transport”, Bull. Sci. Math., 138:2 (2014), 165–198  crossref  mathscinet  zmath  isi  elib
    14. A. V. Kolesnikov, M. Röckner, “On continuity equations in infinite dimensions with non-Gaussian reference measure”, J. Funct. Anal., 266:7 (2014), 4490–4537  crossref  mathscinet  zmath  isi  elib
    15. D. B. Bukin, “On the Kantorovich Problem for Nonlinear Images of the Wiener Measure”, Math. Notes, 100:5 (2016), 660–665  mathnet  crossref  crossref  mathscinet  isi  elib
    16. Backhoff J., Beiglbock M., Lin Y., Zalashko A., “Causal Transport in Discrete Time and Applications”, SIAM J. Optim., 27:4 (2017), 2528–2562  crossref  mathscinet  zmath  isi
    17. 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  mathnet  crossref  mathscinet  zmath
    18. 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  isi
    19. Spantini A., Bigoni D., Marzouk Y., “Inference Via Low-Dimensional Couplings”, J. Mach. Learn. Res., 19 (2018), 66  mathscinet  zmath  isi
    20. 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  mathnet
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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