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Vladimir I. Bogachev

Publications and source records attributed to Vladimir I. Bogachev.

At least 19 recordsLinked to original sources
arXiv2026

Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations

We prove existence of a probability solution to the nonlinear stationary Fokker-Planck-Kolmogorov equation on an infinite dimensional space with a centered Gaussian measure $γ$ with a unit diffusion operator and a drift of the form $-x+v(p,x)$, where $v$ is a bounded mapping with values in the Cameron-Martin space $H$ of $γ$ and $v$ is defined on the space $E\times X$, where is $E$ is the subset of $L^2(γ)$ consisting of probability densities. The equation has the form $L_{b(p,\bullet)} ^*(p\cdot γ)=0$ with $L_{b(p,\bullet)}φ=Δ_H φ+ (b(p,\bullet) , D_{_H}φ)_{_H}$, so that the drift coefficient depends on the unknown solution, which makes the equation nonlinear. This dependence is assumed to satisfy a suitable continuity condition. This result is applied to drifts of Vlasov type defined by means of the convolution of a vector field with the solution. In addition, we consider a more general situation where only the components of $v$ are uniformly bounded and prove the existence of a probability solution under some stronger continuity condition on the drift.

math.AP↗
arXiv2025

Estimates for the distances between solutions to Kolmogorov equations with diffusion matrices of low regularity

We obtain estimates for the weighted $L^1$-norm of the difference of two probability solutions to Kolmogorov equations in terms of the difference of the diffusion matrices and the drifts. Unlike the previously known results, our estimate does not involve Sobolev derivatives of solutions and coefficients. The diffusion matrices are supposed to be non-singular, bounded and satisfy the Dini mean oscillation condition.

math.AP↗
arXiv2024

Compactification of spaces of measures and pseudocompactness

We prove pseudocompactness of a Tychonoff space $X$ and the space $\mathcal{P}(X)$ of Radon probability measures on it with the weak topology under the condition that the Stone-Čech compactification of the space $\mathcal{P}(X)$ is homeomorphic to the space $\mathcal{P}(βX)$ of Radon probability measures on the Stone-Čech compactification of the space~$X$.

math.FA↗
arXiv2022

Kantorovich type topologies on spaces of measures and convergence of barycenters

We study two topologies $τ_{KR}$ and $τ_K$ on the space of measures on a completely regular space generated by Kantorovich--Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich--Rubinshtein topology $τ_{KR}$ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A~sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.

math.PR↗
arXiv2022

Zvonkin's transform and the regularity of solutions to double divergence form elliptic equations

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on~$\mathbb{R}^d$. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix $A$ is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient $b$ is locally integrable to a power $p>d$. We establish new estimates for the $L^p$-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix $A$ satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient.

math.AP↗
arXiv2021

Markov uniqueness and Fokker-Planck-Kolmogorov equations

In this paper we show that Markov uniqueness for symmetric pre-Dirichlet operators $L$ follows from the uniqueness of the corresponding Fokker-Planck-Kolmogorov equation (FPKE). Since in recent years a considerable number of uniqueness results for FPKE's have been achieved, we obtain new Markov uniqueness results in concrete cases. A selection of such will be presented in this paper. They include cases with killing and with degenerate diffusion coefficients.

math.AP↗
arXiv2021

Regularity of solutions to Kolmogorov equations with perturbed drifts

We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation $v$ of the Ornstein--Uhlenbeck operator $L$ possesses a highly integrable density with respect to the Gaussian measure satisfying the non-perturbed equation provided that $v$ is sufficiently integrable. More generally, a similar estimate is proved for solutions to inequalities connected with Markov semigroup generators under the curvature condition $CD(θ,\infty)$. For perturbations from $L^p$ an analog of the Log-Sobolev inequality is obtained. It is also proved in the Gaussian case that the gradient of the density is integrable to all powers. We obtain dimension-free bounds on the density and its gradient, which also covers the infinite-dimensional case.

math.PR↗
arXiv2019

On the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations

We prove a generalization of the known result of Trevisan on the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution is generated by a solution to the corresponding martingale problem. The novelty is that in place of the integrability of the diffusion and drift coefficients $A$ and $b$ with respect to the solution we require the integrability of $(\|A(t,x)\|+|\langle b(t,x),x\rangle |)/(1+|x|^2)$. Therefore, in the case where there are no a priori global integrability conditions the function $\|A(t,x)\|+|\langle b(t,x),x\rangle |$ can be of quadratic growth. Moreover, as a corollary we obtain that under mild conditions on the initial distribution it is sufficient to have the one-sided bound $\langle b(t,x),x\rangle \le C+C|x|^2 \log |x|$ along with $\|A(t,x)\|\le C+C|x|^2 \log |x|$.

math.PR↗
arXiv2018

On $L^1$-estimates for probability solutions to Fokker-Planck-Kolmogorov equations

We prove two new results connected with elliptic Fokker-Planck-Kolmogorov equations with drifts integrable with respect to solutions. The first result answers negatively a long-standing question and shows that a density of a probability measure satisfying the Fokker-Planck-Kolmogorov equation with a drift integrable with respect to this density can fail to belong to the Sobolev class~$W^{1,1}(\mathbb{R}^d)$. There is also a version of this result for densities with respect to Gaussian measures. The second new result gives some positive information about properties of such solutions: the solution density is proved to belong to certain fractional Sobolev classes.

math.PR↗
arXiv2017

A new approach to Nikolskii-Besov classes

We give a new characterization of Nikolskii-Besov classes of functions of fractional smoothness by means of a nonlinear integration by parts formula in the form of a nonlinear inequality. A similar characterization is obtained for Nikolskii-Besov classes with respect to Gaussian measures on finite- and infinite-dimensional spaces.

math.FA↗
arXiv2016

Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy--Landau--Littlewood inequality

We prove that the distribution density of any non-constant polynomial $f(ξ_1,ξ_2,\ldots)$ of degree $d$ in independent standard Gaussian random variables $ξ$ (possibly, in infinitely many variables) always belongs to the Nikol'skii--Besov space $B^{1/d}(\mathbb{R}^1)$ of fractional order $1/d$ (and this order is best possible), and an analogous result holds for polynomial mappings with values in $\mathbb{R}^k$. Our second main result is an upper bound on the total variation distance between two probability measures on $\mathbb{R}^k$ via the Kantorovich distance between them and a suitable Nikol'skii--Besov norm of their difference. As an application we consider the total variation distance between the distributions of two random $k$-dimensional vectors composed of polynomials of degree $d$ in Gaussian random variables and show that this distance is estimated by a fractional power of the Kantorovich distance with an exponent depending only on $d$ and $k$, but not on the number of variables of the considered polynomials.

math.PR↗
arXiv2013

Sobolev functions on infinite-dimensional domains

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.

math.FA↗
arXiv2013

An analytic approach to infinite-dimensional continuity and Fokker-Planck-Kolmogorov equations

We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are presented.

math.PR↗
arXiv2013

Two properties of vectors of quadratic forms in Gaussian random variables

We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting consequences. Our second result gives a characterization of limits in law for sequences of such vectors.

math.PR↗
arXiv2012

Sobolev regularity for the Monge-Ampere equation in the Wiener space

Given the standard Gaussian measure $γ$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot γ$ absolutely continuous with respect to $γ$, we consider the optimal transportation $T(x) = x + \nabla φ(x)$ of $g \cdot γ$ to $γ$. Assume that the function $|\nabla g|^2/g$ is $γ$-integrable. We prove that the function $φ$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 φ) \exp \bigl(\mathcal{L} φ- 1/2 |\nabla φ|^2 \bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $φ$.

math.FA↗
arXiv2008

Mass transport generated by a flow of Gauss maps

Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $μ= \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ν= \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $ν= μ\circ T^{-1}$ and $T = ϕ\cdot {\rm n}$, where $ϕ\colon A \to [0,r]$ is a continuous potential with convex sub-level sets and ${\rm n}$ is the Gauss map of the corresponding level sets of $ϕ$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $ϕ$ the level sets of $ϕ$ are driven by the Gauss curvature flow $\dot{x}(s) = -s^{d-1} \frac{\varrho_1(s {\rm n})}{\varrho_0(x)} K(x) \cdot {\rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.

math.DG↗
arXiv2005

Global Regularity and Bounds for Solutions of Parabolic Equations for Probability Measures

Given a second order parabolic operator $$ Lu(t,x) :=\frac{\partial u(t,x)}{\partial t} + a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x) + b^i(t,x)\partial_{x_i}u(t,x), $$ we consider the weak parabolic equation $L^{*}μ=0$ for Borel probability measures on $(0,1)\times\mathbb{R}^d$. The equation is understood as the equality $$ \int_{(0,1)\times\mathbb{R}^d} Lu dμ=0 $$ for all smooth functions $u$ with compact support in~$(0,1)\times\mathbb{R}^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with~$L$. We show that under broad assumptions $μ$ has the form $μ=\varrho(t,x) dt dx$, where the function $x\mapsto \varrho(t,x)$ is Sobolev, $|\nabla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,τ]\times\mathbb{R}^d$, and $\varrho\in L^p([0,τ]\times\mathbb{R}^d)$ for all $p\in [1,+\infty)$ and $τ<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,τ]\times\mathbb{R}^d$ is given.

math.PR↗