scholarly journals Generalized functions on infinite dimensional spaces and its application to white noise calculus

1989 ◽  
Vol 82 (2) ◽  
pp. 429-464 ◽  
Author(s):  
Yuh-Jia Lee
Keyword(s):  
White Noise ◽  
Generalized Functions ◽  
Infinite Dimensional ◽  
White Noise Calculus

scholarly journals Infinite dimensional rotations and Laplacians in terms of white noise calculus

1992 ◽  
Vol 128 ◽  
pp. 65-93 ◽  
Author(s):  
Takeyuki Hida ◽  
Nobuaki Obata ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Quantum Physics ◽  
Variational Calculus ◽  
Rotation Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian ◽  
White Noise Functionals ◽  
White Noise Calculus

The theory of generalized white noise functionals (white noise calculus) initiated in [2] has been considerably developed in recent years, in particular, toward applications to quantum physics, see e.g. [5], [7] and references cited therein. On the other hand, since H. Yoshizawa [4], [23] discussed an infinite dimensional rotation group to broaden the scope of an investigation of Brownian motion, there have been some attempts to introduce an idea of group theory into the white noise calculus. For example, conformal invariance of Brownian motion with multidimensional parameter space [6], variational calculus of white noise functionals [14], characterization of the Levy Laplacian [17] and so on.

scholarly journals Ito’s formula and Levy’s Laplacian II

1991 ◽  
Vol 123 ◽  
pp. 153-169 ◽  
Author(s):  
Kimiaki Saito
Keyword(s):  
White Noise ◽  
Essential Role ◽  
Beltrami Operator ◽  
Itô’S Formula ◽  
Infinite Dimensional ◽  
Princeton University ◽  
Itô's Formula ◽  
Ito’S Formula ◽  
White Noise Functionals ◽  
White Noise Calculus

The white noise calculus was initiated by T. Hida in 1970 in his Princeton University Mathematical Notes [3]. Recent development of the theory shows that the Laplacian plays an essential role in the analysis in question. Indeed, several kinds of Laplacians should be introduced depending on the choice of the class of white noise functionals to be analysed, as can be seen in [4], [13], [18] and so forth. Among others, we should like to emphasize the importance of the infinite dimensional Laplace-Beltrami operator, Volterra’s Laplacian and Lévy’s Laplacian (See [13], [18] and [20]).

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.

scholarly journals Derivations on white noise functionals

1995 ◽  
Vol 139 ◽  
pp. 21-36 ◽  
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)?

scholarly journals A Hida–Malliavin white noise calculus approach to optimal control

2018 ◽  
Vol 21 (03) ◽  
pp. 1850014
Author(s):  
Nacira Agram ◽  
Bernt Øksendal
Keyword(s):  
Optimal Control ◽  
Diffusion Coefficient ◽  
White Noise ◽  
Backward Stochastic Differential Equation ◽  
Second Order ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Alternative Approach ◽  
Quadratic Optimal Control ◽  
White Noise Calculus

The classical maximum principle for optimal stochastic control states that if a control [Formula: see text] is optimal, then the corresponding Hamiltonian has a maximum at [Formula: see text]. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently, it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first-order derivative was extended to include an extra BSDE for the second-order derivatives. In this paper, we present an alternative approach based on Hida–Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second-order derivatives. The result is illustrated by an example of a constrained linear-quadratic optimal control.

WHITE NOISE ANALYSIS: SOME APPLICATIONS IN COMPLEX SYSTEMS, BIOPHYSICS AND QUANTUM MECHANICS

2012 ◽  
Vol 26 (29) ◽  
pp. 1230014 ◽  
Author(s):  
CHRISTOPHER C. BERNIDO ◽  
M. VICTORIA CARPIO-BERNIDO
Keyword(s):  
Quantum Mechanics ◽  
Probability Density Function ◽  
Complex Systems ◽  
White Noise ◽  
Noise Analysis ◽  
Social Systems ◽  
White Noise Analysis ◽  
The Past ◽  
White Noise Calculus ◽  
With Memory

The white noise calculus originated by T. Hida is presented as a powerful tool in investigating physical and social systems. Combined with Feynman's sum-over-all histories approach, we parameterize paths with memory of the past, and evaluate the corresponding probability density function. We discuss applications of this approach to problems in complex systems and biophysics. Examples in quantum mechanics with boundaries are also given where Markovian paths are considered.

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