Physics – Mathematical Physics
Scientific paper
2005-08-08
in Proceedings of the 8-th International Conference on Path Integrals: From Quantum Information to Cosmology (Prague, Czech Re
Physics
Mathematical Physics
15 pages; some typos removed; some slight change of notation here and there
Scientific paper
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.
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