scholarly journals White noise approach to Gaussian random fields

1990 ◽  
Vol 119 ◽  
pp. 93-106 ◽  
Author(s):  
Ke-Seung Lee
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Integrable Function ◽  
Smooth Boundary ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Main Tool ◽  
Gaussian Random Fields ◽  
Dimensional Euclidean Space ◽  
Square Integrable Function

The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd. Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form(0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du,where W is the Rd-parameter white noise and the kernel F(D, u) is a square integrable function over Rd, and where D is a bounded domain with smooth boundary.

Innovations for Random Fields

1998 ◽  
Vol 01 (04) ◽  
pp. 499-509 ◽  
Author(s):  
Takeyuki Hida ◽  
Si Si
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
White Noise ◽  
Random Fields ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Modern Theory ◽  
White Noise Analysis ◽  
Future Directions ◽  
Probabilistic Structure

There is a famous formula called Lévy's stochastic infinitesimal equation for a stochastic process X(t) expressed in the form [Formula: see text] We propose a generalization of this equation for a random field X(C) indexed by a contour C. Assume that the X(C) is homogeneous in a white noise x, say of degree n, we can then appeal to the classical theory of variational calculus and to the modern theory of white noise analysis in order to discuss the innovation for the X(C) and hence its probabilistic structure. Some of future directions are also mentioned.

Multilevel approximation of Gaussian random fields: Fast simulation

2019 ◽  
Vol 30 (01) ◽  
pp. 181-223 ◽  
Author(s):  
Lukas Herrmann ◽  
Kristin Kirchner ◽  
Christoph Schwab
Keyword(s):  
White Noise ◽  
Fractional Order ◽  
Random Fields ◽  
Computational Cost ◽  
Gaussian Random Fields ◽  
Fast Simulation ◽  
Covariance Operators ◽  
Multilevel Approximation ◽  
Sinc Quadrature ◽  
Consistency Error

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

DIRICHLET FORMS IN TERMS OF WHITE NOISE ANALYSIS I: CONSTRUCTION AND QFT EXAMPLES

1989 ◽  
Vol 01 (02n03) ◽  
pp. 291-312 ◽  
Author(s):  
S. ALBEVERIO ◽  
T. HIDA ◽  
J. POTTHOFF ◽  
M. RÖCKNER ◽  
L. STREIT
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
White Noise ◽  
Random Fields ◽  
Noise Analysis ◽  
Dirichlet Forms ◽  
Quantum Field ◽  
Quantum Fields ◽  
White Noise Analysis ◽  
Nikodym Derivative

Random fields are given in terms of measures which (in general) are singular with respect to that of white noise. However, many such measures can be expressed in terms of white noise through a positive generalized functional acting as a generalized Radon-Nikodym derivative. We give criteria for this to be the case and show that these criteria are fulfilled by Schwinger and Wightman functionals of various nontrivial quantum field theory models. Furthermore a number of closability criteria are given and discussed for the Dirichlet forms associated with positive generalized functionals of white noise. In a second paper we apply these results to the construction of Markov and of quantum fields.

scholarly journals A note on Lévy’s Brownian motion II

1989 ◽  
Vol 114 ◽  
pp. 165-172 ◽  
Author(s):  
Si Si
Keyword(s):  
Brownian Motion ◽  
Random Fields ◽  
Probability Distributions ◽  
Joint Probability ◽  
Variational Calculus ◽  
Normal Derivative ◽  
Gaussian Random Fields ◽  
Normal Direction ◽  
Joint Probability Distributions ◽  
Derivatives Of

The purpose of this paper is to discuss some particular random fields derived from Lévy’s Brownian motion to find its characteristic properties of the joint probability distributions. In [9], special attention was paid to the behaviour of the Brownian motion when the parameter runs along a curve in the parameter space, and with this property the conditional expectation has been obtained when the values are known on the curve.The present paper deals with the variation of the Brownian motion in the normal direction to a given curve, in contrast to the case in [9], where we discussed the properties along the curve. Actually we shall find, in this paper, formulae of the variation with the help of the normal derivative of Brownian motion and observe its singularity. We then discuss partial derivatives of Rd-parameter Lévy’s Brownian motion and make attempt to restrict the parameter to a hypersurface so that we obtain new random fields on that hypersurface. By comparing such derivatives with those of other Gaussian random fields, we can see that the singularity of the new random fields seems to be an interesting characteristic of Lévy’s Brownian motion. Further, we hope that our approach may be thought of as a first step to the variational calculus for Gaussian random fields.

scholarly journals A characterization of the Lévy Laplacian in terms of infinite dimensional rotation groups

1990 ◽  
Vol 118 ◽  
pp. 111-132 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Singular Part ◽  
White Noise Analysis ◽  
Infinite Dimensional ◽  
Functional Derivatives ◽  
Lévy Laplacian ◽  
Rotation Groups

P. Lévy introduced, in his celebrated books [21] and [22], an infinite dimensional Laplacian called the Lévy Laplacian in connection with a number of interesting topics in variational calculus. One of the most significant features of the Lévy Laplacian is observed when it acts on the singular part of the second functional derivatives. For this reason the Lévy Laplacian has become important also in white noise analysis initiated by T. Hida [12]. On the other hand, as was pointed out by H. Yoshizawa [29], infinite dimensional rotation groups are profoundly concerned with the structure of white noise, and therefore, play essential roles in certain problems of stochastic calculus. Motivated by these works, we aim at developing harmonic analysis on infinite dimensional spaces by means of the Lévy Laplacian and infinite dimensional rotation groups.

On the comparison of the distribution of the supremum of random fields represented by stochastic integrals

1989 ◽  
Vol 21 (4) ◽  
pp. 770-780 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Bassan
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Upper Bounds ◽  
Gaussian Random Fields ◽  
Stochastic Integrals ◽  
Response Functions ◽  
Noise Field ◽  
Fixed Radius ◽  
Different Response

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫CpG(P, P′) dW(P′) and U(P) = ∫CpdW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions.The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P).The comparison of the suprema is carried out also, when CP = ℝ2, between fields with different response functions.

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