Design of high-order short-time approximations as a problem of matching the covariance of a Brownian motion

One of the outstanding problems in the numerical discretization of the FeynmanKac 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 ν is equivalent to the problem of constructing discrete-time Gaussian processes that are supported on finitedimensional 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.