The Gauss–Ostrogradsky Formula for the Space of Configurations

Smorodina, N. V.

doi:10.1137/1135102

Cited by 6 publications

(6 citation statements)

References 3 publications

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“…The principal features and main ideas of the method are found in [1,17]. In the case of configuration spaces, the methods of the Malliavin calculus have been employed in [6,12,25,26]. Similar techniques have been used in [8,9,[19][20][21] in the case of measures on locally convex spaces.…”

Section: Surface Measuresmentioning

confidence: 99%

“…Similar techniques have been used in [8,9,[19][20][21] in the case of measures on locally convex spaces. Under more restrictive assumptions, the Gauss-Ostrogradskii formula on a configuration space has been proved in [12,26].…”

Section: Surface Measuresmentioning

confidence: 99%

“…The idea to employ capacities for the study of surface measures is due to Malliavin who has successfully applied it in the Gaussian case. As an application, we obtain a version of the Gauss-Ostrogradskii formula extending a result from [12,26]; formulas of this kind will be useful in the study of boundary value problems on configuration spaces.…”

Section: Introductionmentioning

confidence: 97%

“…In recent years, there has been a growing interest in the analysis and measure theory on configuration spaces of Riemannian manifolds (see, e.g., [3,4,6,12,22,[24][25][26], and the references therein). Configuration spaces provide interesting examples of infinite-dimensional manifolds that possess a relatively simple description and have rich analytic, measure-theoretic, and geometric structures.…”

Section: Introductionmentioning

confidence: 99%

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Surface Measures and Tightness of (r,p)-Capacities on Poisson Space

Богачев

Pugachev

Röckner

2002

Journal of Functional Analysis

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Section: Surface Measuresmentioning

confidence: 99%

Section: Surface Measuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Surface Measures and Tightness of (r,p)-Capacities on Poisson Space

Богачев

Pugachev

Röckner

2002

Journal of Functional Analysis

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“…Assume that Q, = P o Ts -1 = 9,Qo and that r ~ g, is differentiable at zero as a map to LI(Qo). Then P is differentiable along v and Vector fields of differentiability arise naturally in the extensions of the Malliavin calculus to the processes with jumps; see [46,69,72,76,120,161,162,180,182,183,234,235,255,341,413,[519][520][521]595]. In [126], there is a special construction of a vector field on a standard Poisson space such that the corresponding logarithmic derivative coincides with a stochastic integral as in the example above.…”

Section: Differentiability Along Vector Fieldsmentioning

confidence: 99%

Differentiable measures and the Malliavin calculus

Богачев¹

1997

J Math Sci

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IntroductionAmong the most notable events in the nonlinear functional analysis in the past two decades, one can mention the development of the theory of differentiable measures and the creation of the Malliavin calculus. These two theories can be regarded as infinite-dimensional analogs of such classical fields as geometric measure theory, the theory of Sobolev spaces, and the theory of generalized functions.The theory of differentiable measures was suggested by S. V. Fomin in his report at the International Congress of Mathematicians in Moscow in 1966 as an infinite-dimensional substitute for the Sobolev-Schwartz theory of distributions (see [213-215, 36, 37]). Fomin realized that it is natural to consider four spaces, namely, a certain space of functions S, its dual S', a certain space of measures M, and its dual M' in infinite dimensions instead of a pair of spaces (test functions-generalized functions). In the finite-dimensional case, M is identified with S by means of the Lebesgue measure, which allows us to represent measures via their densities. It is the lack of any analogs of the Lebesgue measure that destroys this identification in infinite dimensions. The Fourier transform then acts between S and M and between M' and S'. Thus, a theory of pseudodifferential operators can be developed: the initial Fomin objective was to study infinite-dimensional partial differential equations. However, as often happens with fruitful ideas, the theory of differentiable measures overgrew the initial framework. This theory became an efficient tool in a wide variety of the most diverse applications such as stochastic analysis, quantum field theory, and nonlinear analysis. It is a rapidly developing field which is rich in challenging problems of great importance that provide for better understanding of the nature of infinite-dimensional phenomena. It is worth mentioning that before the pioneering Fomin works similar ideas appeared in several papers by T. Pitcher on the distributions of random processes in functional spaces. Pitcher investigated even more general situations, namely, he studied the differentiability of a family of measures /h = /~ o Tf I generated by a family of transformations Tt of a fixed measure #. Fomin's differentiability corresponds to the case where the measures #t are the shifts #th of the measure # in a linear space. However, the theory constructed for this special case is much richer.In the middle of the seventies, Malliavin [367] suggested an elegant method of proving the smoothness of the transition probabilities of finite-dimensional diffusions. The essence of the method was to consider the transition probabilities Pt as images of the Wiener measure under some nonlinear transformations (generated by stochastic differential equations) and then to apply an integration-by-parts formula on the Wiener space leading to estimates of the generalized derivatives of Pt ensuring the membership in C~ A probabilistic proof of HSrmander's theorem on hypoelliptic second-order operators was given as an applicat...

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Sobolev Capacities of Configurations with Multiple Points in Poisson Space

Pugachev

2004

Mathematical Notes

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The Gauss–Ostrogradsky Formula for the Space of Configurations

Cited by 6 publications

References 3 publications

Surface Measures and Tightness of (r,p)-Capacities on Poisson Space

Surface Measures and Tightness of (r,p)-Capacities on Poisson Space

Differentiable measures and the Malliavin calculus

Sobolev Capacities of Configurations with Multiple Points in Poisson Space

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