Approximations of Integrals Based on Interpolation of Measure

Abstract

This chapter sets forth a method of evaluation of functional integrals w.r.t. Gaussian measure which is based on the formula of the infinitesimal change of measure (or formula of the interpolation of measure) (see [69], [125]-[127]). The method was proposed in connection with difficulties in evaluating integrals over the space of functions X = X ([0, T]) w.r.t. Gaussian measures for functionals of the form exp \( \left\{ {\lambda \int_0^T {V\left( {x\left( \tau \right)} \right)d} \tau } \right\}\) for “large” T, where the term “large” in this case relates to the values of T for which formulae of a given algebraic accuracy degree are inapplicable, including composite ones. Thus, for example, for the integral w.r.t. Gaussian measure with correlation function \(B\left( {t,s} \right) = \frac{1} {2}\exp \left\{ { - \left| {t - s} \right|} \right\}\) of the functional \(f\left( x \right) = \exp \left\{ {0.4i\int_0^T {x\left( \tau \right)d\tau } } \right\}\) the formula of 11-th accuracy degree with real parameters gives no exact digits already for T = 4.0, and for T = 32.0, this formula gives an approximate value which differs from the exact one by one order of magnitude. The method considered in this chapter allows to reduce the evaluation of a functional integral over space of functions defined on segment [0, T] to the evaluation of integrals over space of functions defined on a segment whose length is smaller but is sufficient for the effective applicability of known approximate formulae which are exact for functional polynomials. We shall consider other applications of the formula of measure interpolation at the end of the chapter.