  One of the outstanding problems in the numerical discretization of the
Feynman-Kac formula calls for the design of arbitrary-order short-time
approximations that are constructed in a stable way, yet only require knowledge
of the potential function. In essence, the problem asks for the development of
a functional analogue to the Gauss quadrature technique for one-dimensional
functions. In PRE 69, 056701 (2004), it has been argued that the problem of
designing an approximation of order \nu is equivalent to the problem of
constructing discrete-time Gaussian processes that are supported on
finite-dimensional probability spaces and match certain generalized moments of
the Brownian motion. Since Gaussian processes are uniquely determined by their
covariance matrix, it is tempting to reformulate the moment-matching problem in
terms of the covariance matrix alone. Here, we show how this can be
accomplished.
