Vladimir I. Bogachev

Bio: Vladimir I. Bogachev is an academic researcher from National Research University – Higher School of Economics. The author has contributed to research in topics: Kolmogorov equations & Fokker–Planck equation. The author has an hindex of 32, co-authored 226 publications receiving 3670 citations. Previous affiliations of Vladimir I. Bogachev include Moscow State University & Bielefeld University.

Differentiable Measures and the Malliavin Calculus

Abstract: This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject. Table of Contents: Background material; Sobolev spaces on $\mathbb{R}^n$; Differentiable measures on linear spaces; Some classes of differentiable measures; Subspaces of differentiability of measures; Integration by parts and logarithmic derivatives; Logarithmic gradients; Sobolev classes on infinite dimensional spaces; The Malliavin calculus; Infinite dimensional transformations; Measures on manifolds; Applications; References; Subject index. (Surv/164)

On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions

Abstract: Let A = (aij ) be a matrix-valued Borel mapping on a domain Ω ⊂ R d , let b = (bi ) be a vector field on Ω, and let LA, b ϕ = a ij ∂ x i ∂ xj ϕ + bi ∂ xi ϕ. We study Borel measures μ on Ω that satisfy the elliptic equation LA, b *μ = 0 in the weak sense: ∫ LA, b ϕ dμ = 0 for all ϕ ∈ C 0 ∞ (Ω). We prove that, under mild conditions, μ has a density. If A is locally uniformly nondegenerate, A ∈ H loc p, 1 and b ∈ L loc p for some p > d, then this density belongs to H loc p, 1. Actually, we prove Sobolev regularity for solutions of certain generalized nonlinear elliptic inequalities. Analogous results are obtained in the parabolic case. These results are applied to transition probabilities and invariant measures of diffusion processes.

Fokker-planck-kolmogorov Equations

Abstract: * Stationary Fokker-Planck-Kolmogorov equations* Existence of solutions* Global properties of densities* Uniqueness problems* Associated semigroups* Parabolic Fokker-Planck-Kolmogorov equations* Global parabolic regularity and upper bounds* Parabolic Harnack inequalities and lower bounds* Uniquess of solutions to Fokker-Planck-Kolmogorov equations* The infinite-dimensional case* Bibliography* Subject index

The Monge-Kantorovich problem: achievements, connections, and perspectives

Abstract: This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse applications of these problems in non-linear analysis, probability theory, and differential geometry are discussed. Bibliography: 196 titles.

Generalized Mehler semigroups and applications

Abstract: We construct and study generalized Mehler semigroups (pt)t≧0 and their associated Markov processesM. The construction methods for (pt)t≧0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert spaceH can be extended to some larger Hilbert spaceE, with the embeddingH⊂E being Hilbert-Schmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedXt=C dWt+AXtdt (with possibly unbounded linear operatorsA andC onH) on a suitably chosen larger spaceE. For Gaussian generalized Mehler semigroups (pt)t≧0 with corresponding Markov processM, the associated (non-symmetric) Dirichlet forms (ED(E)) are explicitly calculated and a necessary and sufficient condition for path regularity ofM in terms of (E,D(E)) is proved. Then, using Dirichlet form methods it is shown thatM weakly solves the above stochastic differential equation if the state spaceE is chosen appropriately. Finally, we discuss the differences between these two methods yielding strong resp. weak solutions.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Abstract: Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).