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## Moment measures and stability for Gaussian inequalities

Theory of Stochastic Processes. 2017. Vol. 22. No. 38. P. 47-61.

Let γ be the standard Gaussian measure on Rn and let Pγ be the space of probability measures that are absolutely continuous with respect to γ. We study lower bounds for the functional Fγ(µ) = Ent(µ) − 1 2W2 2 (µ, ν), where µ ∈ Pγ, ν ∈ Pγ, Ent(µ) = R log µ γ dµ is the relative Gaussian entropy, and W2 is the quadratic Kantorovich distance. The minimizers of Fγ are solutions to a dimension-free Gaussian analog of the (real) K¨ahler–Einstein equation. We show that Fγ(µ) is bounded from below under the assumption that the Gaussian Fisher information of ν is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.

Bezhaeva Z., Oseledets V. I., Journal of Dynamical and Control Systems 2013 Vol. 19 No. 2 P. 301-308

Consider a sofic dynamical system. We obtain an explcit formula for the KS-entropy of sofic dynamicsl system of Blackwell's type. ...

Added: August 22, 2013

Kelbert M., Yuri Suhov, Izabella Stuhl, Weighted entropy and optimal portfolios for cautious Kelly investments / Cornell University. Series math "arxiv.org". 2017.

Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes ‘weights’ of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the ...

Added: October 18, 2017

Zelenov G., Theory of Stochastic Processes 2017 Vol. 22 No. 2 P. 79-85

We estimate total variation distances between distributions of polynomials viaL2-norms. ...

Added: November 30, 2019

Karzhemanov I., Zhdanovskiy I., European Journal of Mathematics 2018 Vol. 4 No. 1 P. 326-329

We consider the so-called surjective rational maps. We study how the surjectivity property behaves in families of rational maps. Some (counter) examples are provided and a general result is proved. ...

Added: March 29, 2018

Bufetov A. I., Sevak Mkrtchyan, Maria Scherbina et al., Journal of Statistical Physics 2013 Vol. 152 No. 1 P. 1-14

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles. ...

Added: March 13, 2014

Singapore: World Scientific, 2014

A survey of recent progress and open problems in the theory of Gaussian measures is given. ...

Added: January 3, 2015

Bogachev V., Kolesnikov A., Sobolev regularity for the Monge-Ampere equation in the Wiener space / Cornell University. Series math "arxiv.org". 2011. No. 1110.1822..

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 ...

Added: March 28, 2013

Skripchenko A., Serge Troubetzkoy, Entropy and Complexity of Polygonal Billiards with Spy mirrors / Cornell University. Series math "arxiv.org". 2015. No. 1501.04584.

We prove that a polygonal billiard with one-sided mirrors has zero
topological entropy. In certain cases we show sub exponential and for other
polynomial estimates on the complexity. ...

Added: January 26, 2015

Bogachev V., Kolesnikov A., Теория вероятностей и ее применения 2005 № 50(1) С. 27-51

Показано, что для заданных равномерно выпуклой меры μ на R∞, эквивалентной своему сдвигу на вектор (1,0,0,...), и вероятностной меры ν, абсолютно непрерывной относительно μ, найдется борелевское отображение Т = пространства R∞, переводящее меру μ в v и имеющее вид Т(х) = х + F(x), где F принимает значения в l2. Более того, если мера μ ...

Added: March 23, 2011

Bogachev V., Providence: American Mathematical Society, 1998

This book presents a systematic exposition of the modern theory of Gaussian measures. The basic properties of finite and infinite dimensional Gaussian distributions, including their linear and nonlinear transformations, are discussed. The book is intended for graduate students and researchers in probability theory, mathematical statistics, functional analysis, and mathematical physics. It contains a lot of ...

Added: March 10, 2014

Kolesnikov A., Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 2007 Vol. 18 P. 179-208

We find sufficient conditions for a probability measure $\mu$ to satisfy an inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d \mu} \Bigr) d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|} \Bigr) d \mu + B \int_{\R^d} f^2 d \mu, $$ where $F$ is concave and $c$ (a cost function) is convex. We show ...

Added: March 27, 2013

49606783, V.E. Nazaikinskii, Mathematical notes 2016 Vol. 100 No. 3 P. 421-428

For an arithmetic semigroup (G, ∂), we define entropy as a function on a naturally defined continuous semigroup Ĝ containing G. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (G, ∂). ...

Added: December 10, 2016

Los A., Надежда Вадимовна Вильбоа, Mironkin V., Обозрение прикладной и промышленной математики 2016 Т. 23 № 1 С. 3-16

The tasks associated with finding some information characteristics (entropy, the distribution of m-grams, syllables and phrases), the using of which allows to build the predictive models for the development of natural languages, are considered. The texts in Russian, English, German, French and Georgian languages, as classified by time intervals and styles, are experimental studied. ...

Added: May 8, 2016

Bezhaeva Z., Oseledets V. I., Journal of Dynamical and Control Systems 2013 Vol. 19 No. 4 P. 569-573

Algorithm is given for computation of the Hausdorff dimension of the support of the Erdos measure for a Pisot number. ...

Added: November 11, 2013

V.L. Chernyshev, D.S. Minenkov, V.E. Nazaikinskii, Functional Analysis and Its Applications 2016 Vol. 50 No. 4 P. 291-307

We find the asymptotics of the element counting function for an additive arithmetic semigroup with exponential growth of the counting function of prime generators. ...

Added: October 18, 2015

Popkov Y., Popkov A., Dubnov Y. A., Автоматика и телемеханика 2020 № 7 С. 148-172

A randomized forecasting method based on the generation of ensembles of entropy-optimal forecasting trajectories is developed. The latter are generated by randomized dynamic regression models containing random parameters, measurement noises, and a random input. The probability density functions of random parameters and measurement noises are estimated using real data within the randomized machine learning procedure. ...

Added: October 31, 2020

Remizov I., Моделирование и анализ информационных систем 2015 Vol. 22 No. 3 P. 337-355

A parabolic partial differential equation 𝑢′𝑡(𝑡, 𝑥) = 𝐿𝑢(𝑡, 𝑥) is considered, where 𝐿 is a linear second-order differential operator with time-independent coefficients, which may depend on 𝑥. We assume that the spatial coordinate 𝑥 belongs to a finite- or infinite-dimensional real separable Hilbert space 𝐻. Assuming the existence of a strongly continuous resolving semigroup ...

Added: October 30, 2018

Kosov E., Fractional Calculus and Applied Analysis 2019 Vol. 22 No. 5 P. 1249-1268

We study fractional smoothness of measures on R^k, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption. ...

Added: December 27, 2019

Apenko S.M., Physica A: Statistical Mechanics and its Applications 2012 Vol. 391 No. 1-2 P. 62-77

We present a possible approach to the study of the renormalization group (RG) flow based
entirely on the information theory. The average information loss under a single step of
Wilsonian RG transformation is evaluated as a conditional entropy of the fast variables,
which are integrated out, when the slow ones are held fixed. Its positivity results in the
monotonic ...

Added: October 23, 2014

Bogachev V., Kolesnikov A., Medvedev K. V., Математический сборник 2005 Т. 196 № 3 С. 3-30

Получено новое тождество для энтропии нелинейного образа меры на Rn, дающее известное неравенство Талаграна. Исследованы треугольные отображения в Rn и R∞, т.е. отображения T, у которых i-я координатная функция T
i зависит только от переменных x 1,…, xi. С помощью этих отображений дано положительное решение известной открытой проблемы о представимости всякой вероятностной меры ν, абсолютно непрерывной ...

Added: March 26, 2013

49606783, Russian Journal of Mathematical Physics 2017 Vol. 24 No. 3 P. 354-372

In the paper, a new construction of the theory of partitions of integers is proposed.
The author defines entropy as the natural logarithm of the number of partitions of a number
M into natural summands with repetitions allowed p(M) and repetitions forbidden q(M).
The passage from ln p(M) to lnq(M) through the mesoscopic values M → 0 is ...

Added: November 17, 2018

Popkov Y., Dubnov Y. A., Zeev Volkovich et al., Entropy 2017 Vol. 19(4) No. 178 P. 1-14

A proposal for a new method of classification of objects of various nature, named “2”-soft classification, which allows for referring objects to one of two types with optimal entropy probability for available collection of learning data with consideration of additive errors therein. A decision rule of randomized parameters and probability density function (PDF) is formed, ...

Added: May 26, 2017

Bogachev V., A. Yu. Pilipenko, A.V. Shaposhnikov, Journal of Mathematical Analysis and Applications 2014 Vol. 419 No. 2 P. 1023-1044

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space. ...

Added: January 3, 2015

Bufetov A. I., Geometric and Functional Analysis 2012 Vol. 22 No. 4 P. 938-975

Vershik and Kerov conjectured in 1985 that dimensions of irreducible representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The statement of the Vershik-Kerov conjecture can be seen as an analogue of the Shannon-McMillan-Breiman Theorem for the non-stationary ...

Added: October 18, 2012