Papers by Vladimir Bogachev
Mathematical Surveys and Monographs, 1998
Mathematical Surveys and Monographs, 1998
Mathematical Surveys and Monographs, 2010
Math Notes Engl Tr, 1986
ABSTRACT Let α∈(0,2], X be a locally convex space and X * its dual. For every f∈X * and every mea... more ABSTRACT Let α∈(0,2], X be a locally convex space and X * its dual. For every f∈X * and every measure Γ≥0 on X with bounded support let Q(α,Γ,f)=tg(πα/2)∫f|f| α-1 dΓ for α≠1 and -2π -1 ∫fln|f|dΓ for α=1· The author introduces the concept of asymmetry index β(μ) of a stable probability μ on X, with exponent α. If X=R n , β(μ) is the infimum of all β≥0 for which the characteristic function of μ may be represented as (1)ϕ μ (f)=exp(if(a)-∫|f| α dΓ+iβQ(α,Γ,f)), with an a∈X and some Γ. For a general X, β(μ) is the supremum of all β(μ∘T -1 ) with linear continuous T:X→R n , n=1,2,···. We have β(μ)∈[0,1]· An open problem is: does β(μ) change if we restrict to n=1? The author proves that β(μ∘T -1 )≤β(μ) for a linear continuous T, with = if kerT=0, β(⊗ n≥1 μ n )=sup n β(μ n ) and that β(μ) is the absolute value of the ”usual coefficient” for X=R and for the distribution on the function space of a homogeneous stable process with independent increments. Every stable μ equals μ 1 *ν, where μ 1 is the symmetrisation of μ transported by a c.1 and ν is stable with β(ν)=1 if β(μ)>0 and degenerated if β(μ)=0. c is expressed in terms of β(μ) and is extremal among all for which such a representation, without supposing ν stable, exists. If for all stable probabilities on X the representation (1) is possible with β=1 (particularily if X is quasicomplete), then the set of all β≥0 for which this is possible is [β(μ),1]·
Doklady Mathematics, 2015
Sbornik Mathematics, Jun 30, 1998
Let \mu be a centred Gaussian measure in a linear space X with Cameron-Martin space H, let q be a... more Let \mu be a centred Gaussian measure in a linear space X with Cameron-Martin space H, let q be a \mu-measurable seminorm, and let Q be a \mu-measurable second-order polynomial. We show that it is sufficient for the existence of the limit \lim _{\varepsilon \to 0}\mathsf E(\exp Q\vert q\leqslant \varepsilon), where E is the expectation with respect to \mu, that the second derivative D_{\!H}^{\,2}Q of the function Q be a nuclear operator on H. This condition is also necessary for the existence of the above-mentioned limit for all seminorms q. The problem under discussion can be reformulated as follows: study \lim _{\varepsilon \to 0}\nu (q\leqslant \varepsilon )/\mu (q\leqslant \varepsilon ) for Gaussian measures \nu equivalent to \mu.
Various results are described on invariant measures of dξ t =dw t +B(ξ t )dt. The reversibility o... more Various results are described on invariant measures of dξ t =dw t +B(ξ t )dt. The reversibility of this infinite-dimensional diffusion is discussed using Fomin’s differentiation and vector logarithmic derivatives of measures. The main result of this article which anticipates and summarises in short results in a paper of the author with M. Roeckner is an example of a nondegenerate Gaussian diffusion in a Hilbert space for which an invariant probability is not unique.
This is the English translation of the original Russian edition from 1997, see the review Zbl 088... more This is the English translation of the original Russian edition from 1997, see the review Zbl 0883.60032.
From the text: Infinite-dimensional integration-by-parts formulae are given and some properties o... more From the text: Infinite-dimensional integration-by-parts formulae are given and some properties of logarithmic derivatives in the framework of the theory of differentiable measures are investigated.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
In this paper a bounded continuous vector field on a Banach space is constructed such that the co... more In this paper a bounded continuous vector field on a Banach space is constructed such that the corresponding ordinary differential equation is uniquely solvable, while the stochastic differential equation with this field as drift has no solutions. The second example gives a solvable stochastic equation for which the deterministic one, with the same vector field, has no solution.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
We consider oscillatory integrals J(t)=∫exp(itF(x))μ(dx), where F is a smooth function (in some s... more We consider oscillatory integrals J(t)=∫exp(itF(x))μ(dx), where F is a smooth function (in some sense) on an infinite-dimensional measurable space (X,μ). In particular we study the case when F is a polynomial or integral-type functional and μ is a smooth (for example, Gauss) measure or the distribution of a diffusion process. Connections with the Malliavin calculus are discussed.
Теория вероятностей и ее применения, 2010
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Papers by Vladimir Bogachev