Analytic properties of infinite-dimensional distributions

Богачев, В. И.; Smolyanov, O. G.

doi:10.1070/rm1990v045n03abeh002364

Cited by 64 publications

(77 citation statements)

References 207 publications

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“…Earlier S. V. Fomin defined partial logarithmic derivatives in his theory of differentiable measures (see [9,14] for surveys and references). Clearly, the existence of a vector logarithmic derivative β µ H implies partial differentiability in Fomin's sense along all vectors j H (k), k ∈ X * (and the equality…”

Section: The Infinite-dimensional Casementioning

confidence: 99%

On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Albeverio

Богачев

Röckner

1999

Comm. Pure Appl. Math.

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Section: The Infinite-dimensional Casementioning

confidence: 99%

On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Albeverio

Богачев

Röckner

1999

Comm. Pure Appl. Math.

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“…In infinite dimensions there exists an exact analogue of the notion of a measure with a smooth density: this is a measure differentiable in Fomin's sense (see [13,25] for references). As opposed to the finite-dimensional situation, however, the study of such measures cannot be reduced to that of Gaussian measures in the sense that, as shown in [6] (see also [13]), there exist smooth measures on infinite-dimensional spaces mutually singular with respect to all Gaussian measures.…”

Section: Moreover U=umentioning

confidence: 99%

Absolutely Continuous Flows Generated by Sobolev Class Vector Fields in Finite and Infinite Dimensions

Богачев

Mayer-Wolf

1999

Journal of Functional Analysis

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We prove the existence of the global flow [U t ] generated by a vector field A from a Sobolev class W 1, 1 (+) on a finite-or infinite-dimensional space X with a measure +, provided + is sufficiently smooth and that a {A and |$ + A| (where $ + A is the divergence with respect to +) are exponentially integrable. In addition, the measure + is shown to be quasi-invariant under [U t ]. In the case X=R n and += p dx, where p # W 1, 1 loc (R n ) is a locally uniformly positive probability density, a sufficient condition is exp(c &{A&)+exp(c |(A, ({pÂ p))| ) # L 1 (+) for all c. In the infinitedimensional case we get analogous results for measures differentiable along sufficiently many directions. Examples of measures which fit our framework, important for applications, are symmetric invariant measures of infinite-dimensional diffusions and Gibbs measures. Typically, in both cases such measures are essentially nonGaussian. Our result in infinite dimensions significantly extends previously studied cases where + was a Gaussian measure. Finally, we study flows generated by vector fields whose values are not necessarily in the Cameron Martin space. 1999Academic Press

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“…In what follows, we refer to the first system (as well as to its classical counterpart) as the open system and to the second as the environment, respectively. These two systems form a composite system, whose Hilbert space is the Hilbert tensor product [3] …”

Section: States Of Open Quantum Systemsmentioning

confidence: 99%

“…Moreover, we assume that the quantum systems under consideration are obtained by the Schrödinger quantization [3] of classical Hamiltonian systems.…”

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Feynman, Wigner, and Hamiltonian structures describing the dynamics of open quantum systems

2014

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This paper discusses several methods for describing the dynamics of open quantum systems, where the environment of the open system is infinite-dimensional. These are purifications, phase space forms, master equation and liouville equation forms. The main contribution is in using Feynman-Kac formalisms to describe the infinite-demsional components.This paper discusses several approaches for describing the dynamics of open quantum systems. Open quantum systems play an important role in modelling physical systems coupled to their environment and, in particular, for the emerging field of quantum feedback control theory (see [12]). Thus, in studying coherent quantum feedback, models consisting of a quantum system related to quantum control system, so that each of these systems turns out to be open, are considered.Generally the master equation is presented in the theoretical physics literature as the central description of an open quantum system. In practice, however, it is these solutions that are important for potential applications, rather than the master equations themselves. Our goal is to solve the exact master equations, which describe the reduced dynamics of subsystems of certain large systems generated by the dynamics of these large systems: these master equation arise from a number of different approaches which we will consider.In fact, we examine four approaches for describing subsystems dynamics, and in each case we exploit Feynman type formulas (see [1,2]). Moreover, we assume that the quantum systems under consideration are obtained by the Schrödinger quantization [3] of classical Hamiltonian systems.1. Our first approach is based on a representation of mixed states as random pure ones, where the dynamics of a subsystem of the isolated quantum

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Analytic properties of infinite-dimensional distributions

Cited by 64 publications

References 207 publications

On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Absolutely Continuous Flows Generated by Sobolev Class Vector Fields in Finite and Infinite Dimensions

Feynman, Wigner, and Hamiltonian structures describing the dynamics of open quantum systems

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