Positive Generalized Functions on Infinite Dimensional Spaces

1993 ◽  
pp. 225-234 ◽  
Author(s):  
Yuh-Jia Lee
Keyword(s):  
Generalized Functions ◽  
Infinite Dimensional

Infinite degrees of freedom Weyl representation: Characterization and application

2018 ◽  
Vol 21 (01) ◽  
pp. 1850002
Author(s):  
Abdessatar Barhoumi ◽  
Bilel Kacem Ben Ammou ◽  
Hafedh Rguigui
Keyword(s):  
Degrees Of Freedom ◽  
Explicit Construction ◽  
Stochastic Calculus ◽  
Generalized Functions ◽  
Academic Press ◽  
Quantum Stochastic Calculus ◽  
Fock Representation ◽  
Infinite Dimensional ◽  
Weyl Representation ◽  
Nuclear Spaces

By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.

Stochastic Differential Equations in White Noise Space

1998 ◽  
Vol 01 (04) ◽  
pp. 611-632 ◽  
Author(s):  
H.-H. Kuo ◽  
J. Xiong
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Generalized Functions ◽  
Two Dimensional ◽  
Stochastic Fluctuation ◽  
Infinite Dimensional ◽  
Distribution Laws ◽  
White Noise Space

We study infinite dimensional stochastic differential equations taking values in a white noise space. We show that under certain assumptions the distribution laws of the solution of such an equation induce generalized functions. The white noise integral equation satisfied by these generalized functions is derived. We apply the results to study the stochastic fluctuation of a two-dimensional neuron.

LOCALLY CONVEX SPACES OF VECTOR-VALUED DISTRIBUTIONS WITH MULTIPLICATIVE STRUCTURES

2002 ◽  
Vol 05 (04) ◽  
pp. 483-502 ◽  
Author(s):  
A. YU. KHRENNIKOV ◽  
V. M. SHELKOVICH ◽  
O. G. SMOLYANOV
Keyword(s):  
Linear Space ◽  
Commutative Algebra ◽  
Linear Span ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Functional Interpretation ◽  
Infinite Dimensional ◽  
Space Of Distributions ◽  
Vector Valued ◽  
Locally Convex

We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(fn(x)) finite from the left (i.e. fn(x)=0 for n<n0(f*)) whose components fn(x) belong to the linear span [Formula: see text] generated by the distributions δ(m-1)(x-ck), P((x-ck)-m), xm-1, where m=1, 2, …, ck ∈ ℝ, k = 1, …, s. The space of distributions [Formula: see text] can be realized as a subspace in [Formula: see text] This linear space [Formula: see text] has the structure of an associative and commutative algebra containing a unity element and free of zero divizors. The Schwartz counterexample does not hold in the algebra [Formula: see text]. Unlike the Colombeau algebra, whose elements have no explicit functional interpretation, elements of the algebra [Formula: see text] are infinite-dimensional Schwartz vector-valued distributions. This construction can be considered as a next step and a "model" on the way of constructing a nonlinear theory of distributions similar to that developed by L. Schwartz. The obtained results can be considerably generalized.

scholarly journals Generalized functions in infinite-dimensional analysis

1998 ◽  
Vol 28 (2) ◽  
pp. 213-260 ◽  
Author(s):  
Yuri G. Kondratiev ◽  
Ludwig Streit ◽  
Werner Westerkamp ◽  
Jia-an Yan
Keyword(s):  
Dimensional Analysis ◽  
Generalized Functions ◽  
Infinite Dimensional ◽  
Infinite Dimensional Analysis
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