Random series and discrete path integral methods: The Lévy-Ciesielski implementation.
We perform a thorough analysis of the relationship between discrete and series representation path integral methods, which are the main numerical techniques used in connection with the Feynman-Kaç formula. First, an interpretation of the so-called standard discrete path integral methods is derived by direct discretization of the Feynman-Kaç formula. Second, we consider a particular random series technique based upon the Lévy-Ciesielski representation of the Brownian bridge and analyze its main implementations, namely the primitive, the partial averaging, and the reweighted versions. It is shown that the n=2(k)-1 subsequence of each of these methods can also be interpreted as a discrete path integral method with appropriate short-time approximations. We therefore establish a direct connection between the discrete and the random series approaches. In the end, we give sharp estimates on the rates of convergence of the partial averaging and the reweighted Lévy-Ciesielski random series approach for sufficiently smooth potentials. The asymptotic rates of convergence are found to be O(1/n(2)), in agreement with the rates of convergence of the best standard discrete path integral techniques.