Approximation of probability measures on a Hilbert space by convex combinations of dirac measures

1979 ◽  
Vol 26 (2) ◽  
pp. 615-618
Author(s):  
I. V. Kozin
Keyword(s):  
Hilbert Space ◽  
Probability Measures ◽  
Approximation Of Probability Measures ◽  
Dirac Measures

scholarly journals Operator semistable probability measures on a Hilbert space: Corrigenda

1980 ◽  
Vol 22 (3) ◽  
pp. 479-480
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Probability Measures ◽  
Semistable Probability

A faulty typescript of [2] was, regrettably, submitted. The following changes should be made:Page 398, line 7: replace A є G with A є L.Page 398, line 12: replace A є G with A є L.

scholarly journals Operator semistable probability measures on a Hilbert space

1980 ◽  
Vol 22 (3) ◽  
pp. 397-406 ◽  
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Real Separable Hilbert Space ◽  
Semistable Probability

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

scholarly journals Control problem on space of random variables and master equation

2019 ◽  
Vol 25 ◽  
pp. 10
Author(s):  
Alain Bensoussan ◽  
Sheung Chi Phillip Yam
Keyword(s):  
Hilbert Space ◽  
Control Problem ◽  
Master Equation ◽  
Mean Field ◽  
Random Variables ◽  
Probability Measures ◽  
Control Problems ◽  
Field Type ◽  
Collège De France ◽  
Mean Field Type Control

In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. We see that the control problem can then be related to a mean field type control problem. We explore here a suggestion of Lions in (Lectures at College de France, http://www.college-de-france.fr) and (Seminar at College de France). Mean field type control problems are control problems in which functionals depend on probability measures of the underlying controlled process. Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643] considered this type of problem in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient; their work provides a completely rigorous treatment, but it is quite intricate, because metric spaces are not vector spaces. The approach suggested by Lions overcomes this difficulty. Nevertheless, our present proposed approach also benefits from the useful concept of L-derivatives as introduced in a recent interesting treatise of Carmona and Delarue [Probabilistic Theory of Mean Field Games with Applications. Springer Verlag (2017)]. We also consider Bellman equation and the Master equation of mean field type control. We provide also some extension of the results of Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643].

PROBABILITY STRUCTURES FOR QUANTUM STATE SPACES

1995 ◽  
Vol 07 (07) ◽  
pp. 1105-1121 ◽  
Author(s):  
PAUL BUSCH ◽  
GIANNI CASSINELLI ◽  
PEKKA J. LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Quantum State ◽  
Positive Operator ◽  
Trace Class ◽  
Measurable Space ◽  
Probability Measures ◽  
State Spaces ◽  
Trace Class Operators ◽  
Positive Operator Valued Measure

The theme of this paper is to represent the states of a quantum system by means of probability measures. We fix a positive operator valued measure E on a measurable space (Ω, ℬ(Ω)) acting in a Hilbert space ℋ, and we study the properties of the mapping that it induces from the set of trace class operators on ℋ to the set of measures on (Ω, ℬ(Ω)). In particular, the injectivity and the surjectivity of this map are characterised in terms of the properties of E.

scholarly journals Ergodic sequences of probability measures on commutative hypergroups

2004 ◽  
Vol 2004 (7) ◽  
pp. 335-343 ◽  
Author(s):  
Liliana Pavel
Keyword(s):  
Hilbert Space ◽  
Probability Measures ◽  
Commutative Hypergroup

We study conditions on a sequence of probability measures{μn}non a commutative hypergroupK, which ensure that, for any representationπofKon a Hilbert spaceℋπand for anyξ∈ℋπ,(∫Kπx(ξ)dμn(x))nconverges to aπ-invariant member ofℋπ.

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