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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 1, Pages 27–51 (Mi tvp157)

Nonlinear transformations of convex measures

V. I. Bogachev, A. V. Kolesnikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Given a uniformly convex measure $\mu$ on $R^\infty$ that is equivalent to its translation to the vector $(1,0,0,\ldots)$ and a probability measure $\nu$ that is absolutely continuous with respect to $\mu$, we show that there is a Borel mapping $T=(T_k)_{k=1}^\infty$ of $R^\infty$ transforming $\mu$ into $\nu$ and having the form $T(x)=x+F(x)$, where $F$ has values in $l^2$. Moreover, if $\mu$ is a product-measure, then $T$ can be chosen triangular in the sense that each component $T_k$ is a function of $x_1,…,x_k$. In addition, for any uniformly convex measure $\mu$ on $R^\infty$ and any probability measure $\nu$ with finite entropy $\textrm{ent}_\mu(\nu)$ with respect to $\mu$, the canonical triangular mapping $T=I+F$ transforming $\mu$ into $\nu$ satisfies the inequality $\|F\|_{L^2(\mu,l^2)}^2\le C(\mu)\textrm{ent}_\mu (\nu)$. Several inverse assertions are proved. Our results apply, in particular, to the standard Gaussian product-measure. As an application we obtain a new sufficient condition for the absolute continuity of a nonlinear image of a convex measure and the membership of the corresponding Radon–Nikodym derivative in the class $L\log L$.

Keywords: convex measure, Gaussian measure, product-measure, Cameron–Martin space, absolute continuity, triangular mapping.

DOI: https://doi.org/10.4213/tvp157

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English version:
Theory of Probability and its Applications, 2006, 50:1, 34–52

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Citation: V. I. Bogachev, A. V. Kolesnikov, “Nonlinear transformations of convex measures”, Teor. Veroyatnost. i Primenen., 50:1 (2005), 27–51; Theory Probab. Appl., 50:1 (2006), 34–52

Citation in format AMSBIB
\Bibitem{BogKol05} \by V.~I.~Bogachev, A.~V.~Kolesnikov \paper Nonlinear transformations of convex measures \jour Teor. Veroyatnost. i Primenen. \yr 2005 \vol 50 \issue 1 \pages 27--51 \mathnet{http://mi.mathnet.ru/tvp157} \crossref{https://doi.org/10.4213/tvp157} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2222736} \zmath{https://zbmath.org/?q=an:1091.28009} \elib{https://elibrary.ru/item.asp?id=9153104} \transl \jour Theory Probab. Appl. \yr 2006 \vol 50 \issue 1 \pages 34--52 \crossref{https://doi.org/10.1137/S0040585X97981457} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000236850700003} 

• http://mi.mathnet.ru/eng/tvp157
• https://doi.org/10.4213/tvp157
• http://mi.mathnet.ru/eng/tvp/v50/i1/p27

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This publication is cited in the following articles:
1. V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev, “Triangular transformations of measures”, Sb. Math., 196:3 (2005), 309–335
2. 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
3. Kirill V. Medvedev, “Certain properties of triangular transformations of measures”, Theory Stoch. Process., 14(30):1 (2008), 95–99
4. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890
5. Bogachev V.I. Kolesnikov A.V., “Sobolev Regularity for the Monge-Ampere Equation in the Wiener Space”, Kyoto J. Math., 53:4 (2013), 713–738
6. Kolesnikov A.V., Roeckner M., “On Continuity Equations in Infinite Dimensions with Non-Gaussian Reference Measure”, J. Funct. Anal., 266:7 (2014), 4490–4537
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