Vladimir Bogachov

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Speaker

Vladimir Igorevich Bogachev is a plenary speaker, professor of mathematics at the Moscow State University and Moscow Higher School of Economics.

Vladimir received his Ph.D. degree from Moscow State University in 1986 under the direction of professor Smolyanov, and later, in 1991, his Dr. Sc. degree also from MSU. Since 1996 he is Professor of Math. Department at MSU. His main research interests cover much of Measure theory, Probability theory and Stochastic processes, infinite dimensional analysis. Professor Bogachev is the author of more then 170 research papers and of 11 monographs including Gaussian measures, American Math. Soc. (1998),  Measure theory, 1, 2, Springer (2007), Differentiable measures and the Malliavin calculus, American Math. Soc. (2010). He has been an invited speaker at various international conferences on different branches of Analysis and Probability. He received Gold Medal from Russian Academy of Science (1990) and Award of the The Japan Society for Promotion of Science

Lecture

Distributions of polynomials in many variables and Nikolskii-Besov spaces

We shall discuss recent progress in the study of spaces of fractional regularity and interesting connections of this classical area with another topic that has also been actively developing in the past decade: distributions of polynomials on spaces of high or even infinite dimension equipped with measures. One simple instance of such connections is the fact that the distribution of an arbitrary non-constant polynomial of degree [math]d[/math] in finitely many or infinitely many Gaussian random variables belongs to the Nikolskii-Besov class of fractional order [math]1/d[/math] independently of the number of variables of the polynomial. The method of proving this rather unexpected and nontrivial fact is based on a new characterization of Besov spaces via a certain nonlinear integration by parts formula. The classical Sobolev space of functions [math]f[/math] in [math]L^p[/math] with first order generalized partial derivatives in [math]L^p[/math] is equivalently described by the estimate

$$ \int \varphi' f\, dx \le C \|\varphi\|_q $$

for all test functions [math]\varphi[/math], where [math]q=p/(p-1)[/math]. For [math]p=1[/math] and [math]q=\infty[/math] this estimate is equivalent to the inclusion of [math]f[/math] into the space [math]BV[/math] of functions of bounded variation. It turns out that the Nikolskii-Besov space defined through the integral modulus of continuity

$$ \int |f(x+h)-f(x)|\, dx \le C |h|^\alpha $$

can be also characterized by the estimate

$$ \int \varphi' f\, dx \le C \|\varphi\|_\infty^{\alpha}\|\varphi'\|_\infty^{1-\alpha}. $$

Similar properties can be considered for other measures in place of Lebesgue measure, for example, for the Gaussian measure, which leads to important infinite-dimensional generalizations. On multidimensional spaces and manifolds, one can also introduce such properties by using derivatives along vector fields in place of partial derivatives (then divergences in place of derivatives appear on the right-hand side). Yet another way of expressing such properties is connected with semigroups such as the heat semigroup and the Ornstein-Uhlenbeck semigroup. The lecture will give a concise introduction to this new direction of research including formulations of open problems. All necessary notions will be introduced and the presentation will be relatively elementary.