The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

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BV functions in a Gelfand triple for differentiable measure and its applications

Forum Mathematicum ◽

10.1515/forum-2012-0137 ◽

2015 ◽

Vol 27 (3) ◽

Cited By ~ 6

Author(s):

Michael Röckner ◽

Rongchan Zhu ◽

Xiangchan Zhu

Keyword(s):

Dirichlet Form ◽

Adjoint Operator ◽

Stochastic Quantization ◽

Bv Functions ◽

Form Theory ◽

Differentiable Measure ◽

Definition Of ◽

Non Gaussian ◽

Gelfand Triple ◽

Quantization Problem

AbstractIn this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [Ann. Probab. 40 (2012), 1759–1794], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator

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Fidelity, sub-fidelity, super-fidelity and their preservers

International Journal of Quantum Information ◽

10.1142/s0219749915500276 ◽

2015 ◽

Vol 13 (03) ◽

pp. 1550027

Author(s):

Li Wang ◽

Jinchuan Hou ◽

Kan He

Keyword(s):

Hilbert Space ◽

Unitary Operator ◽

Convex Set ◽

Quantum Systems ◽

Complex Hilbert Space ◽

Quantum States ◽

Infinite Dimensional

Sub- and super-fidelity describe respectively the lower and super bound of fidelity of quantum states. In this paper, we obtain several properties of sub- and super-fidelity for both finite- and infinite-dimensional quantum systems. Furthermore, let H be a separable complex Hilbert space and ϕ : 𝒮(H) → 𝒮(H) a map, where 𝒮(H) denotes the convex set of all states on H. We show that, if dim H < ∞, or, if dim H = ∞ and ϕ is surjective, then the following statements are equivalent: (1) ϕ preserves the super-fidelity; (2) ϕ preserves the fidelity; (3) ϕ preserves the sub-fidelity; (4) there exists a unitary or an anti-unitary operator U on H such that ϕ(ρ) = UρU† for all ρ ∈ 𝒮(H).

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Local affine selections of convex set-valued functions

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2062 ◽

2019 ◽

Vol 26 (4) ◽

pp. 621-624

Author(s):

Szymon Wąsowicz

Keyword(s):

Banach Spaces ◽

Main Part ◽

Convex Set ◽

Dimensional Case ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Finite Dimensional Case

Abstract It is well known that not every convex set-valued function admits an affine selection. One could ask whether there exists at least one local affine selection. The answer is positive in the finite-dimensional case. The main part of the note comprises two examples of the non-existence of local affine selections of convex set-valued functions defined on certain infinite-dimensional Banach spaces.

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Derivations on white noise functionals

Nagoya Mathematical Journal ◽

10.1017/s0027763000005286 ◽

1995 ◽

Vol 139 ◽

pp. 21-36 ◽

Cited By ~ 2

Author(s):

Nobuaki Obata

Keyword(s):

White Noise ◽

Topological Algebra ◽

Fundamental Question ◽

Special Choice ◽

Pointwise Multiplication ◽

Infinite Dimensional ◽

White Noise Functionals ◽

White Noise Calculus ◽

Gelfand Triple ◽

Gaussian Space

The Gaussian space (E*, μ) is a natural infinite dimensional analogue of Euclidean space with Lebesgue measure and a special choice of a Gelfand triple gives a fundamental framework of white noise calculus [2] as distribution theory on Gaussian space. It is proved in Kubo-Takenaka [7] that (E) is a topological algebra under pointwise multiplication. The main purpose of this paper is to answer the fundamental question: what are the derivations on the algebra (E)?

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