Accurate evaluation of Wiener integrals

Alexandre Joel Chorin
1973 Mathematics of Computation  
A new quadrature formula for an important class of Wiener integrals is presented, in which the Wiener integrals are approximated by «-fold integrals with an error Oin~2). The resulting «-fold integrals can then be approximated by ordinary finite sums of remarkably simple structure. An example is given. Introduction. Wiener integrals in function space play a major role in a number of applications in physics and in probability theory, see e.g. [1], [6], [7], [9]. A number of remarkable results
more » ... e been obtained concerning the approximation of these integrals by finite-dimensional integrals (see in particular Cameron [2], as well as [8], [10], and [14] ). The resulting n-fold integrals are, in general, difficult to evaluate with any accuracy, and as a consequence the approximation formulas are not of significant practical use. The aim of this paper is to present a new approximation for Wiener integrals accurate enough and simple enough to be of practical interest. Some of the elegant generality of Cameron's work may be lost, but the method is applicable to many functionals which appear in physics, and will furthermore afford an intuitive grasp of the relation between ordinary quadrature and quadrature in a function space. The two main ideas in the approximation method are the following: the Wiener paths are carefully interpolated by a certain family of parabolas, in such a way that all the moments are exactly reproduced; and nonlinear functionals are expanded in a certain Taylor series, with the quadrature formula adjusted so that the first two groups of terms are well approximated. Outline of Goal and Method. Let C be the space of continuous real functions x(i) defined on 0 ^ t ^ 1, with x(0) = 0, and endowed with the Wiener measure W. Let F[x] be a functional on C; our aim is to evaluate J = j F[x] dW; J c we shall construct approximation formulas of the form m \ F[x] dW = x~n/2 [ FniUl, u2, ■■■ , un) (l) Jc J R» •exp(-«i -u2 -• • ■ -un) dux du2 • ■ ■ dun + Oin'2)
doi:10.1090/s0025-5718-1973-0329205-7 fatcat:ioddvskgvvhn5azwtdy7nhoh3m