Computation of conditional Wiener integrals by the composite approximation formulas with weight
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.
family of approximate formulas with weight
scientific article; zbMATH DE number 4126560