Computation of conditional Wiener integrals by the composite approximation formulas with weight
type: Item
instance of: scholarly article
title: Computation of conditional Wiener integrals by the composite approximation formulas with weight
zbMATH Open document ID: 0687.65016
DOI: 10.1016/0377-0427(90)90194-5
author: K. Appert
published in: Journal of Computational and Applied Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The authors study the approximate calculation of the conditional Wiener integral \(\int_{C}P[x]F[x]d_ wx\) considering it a Lebesgue integral with respect to a conditional Wiener measure. The measure \(d_ wx\) is a Gaussian measure suitably defined on the space \(c=\{x(t)\in C[0,1]\); \(x(0)=x(1)=0\}\). F[x] is a real functional defined on C, \[ P[x]=\exp \{\int^{1}_{0}(p(t)x^ 2(t)+q(t)x(t))dt\},\quad p,q\in C[0,1], \] is a weight functional. The authors derive a family of approximate formulas with weight, improving a previous result of the first, the third and the fourth author [Soobshch. Ob''edin. Inst. Yad. Issled. Dubna P-11-83-867, 14 p. (1983; Zbl 0543.65009)] concerning the functional integral with respect to a Gaussian measure without weight. These formulas, whose expression is too complicated to be referred here, depend on a natural parameter m and are exact for every polynomial functional of degree less or equal than \(2m+1\). The authors prove also the convergence of the method and obtain an estimate of the remainder. These calculations are connected with problems of quantum mechanics, as is shown by some examples in the paper, where the method is claimed to be more efficient than the Monte Carlo method usually employed.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 4126560
zbMATH Keywords: conditional Wiener integral
zbMATH Keywords: conditional Wiener measure
zbMATH Keywords: family of approximate formulas with weight
zbMATH Keywords: convergence
zbMATH Keywords: estimate of the remainder
zbMATH Keywords: Monte Carlo method
MaRDI profile type: Publication
full work available at URL: 0377-0427(90)90194-5
OpenAlex ID: W2009301243
cites work: Q3702381
cites work: Integration in Functional Spaces and its Applications in Quantum Physics
cites work: Q3910965
cites work: On the deterministic computation of functional integrals in application to quantum mechanical problems
cites work: Q3332151
cites work: Q4204913
description: scientific article; zbMATH DE number 4126560