On the approximation of Feynman–Kac path integrals

A general framework is proposed for the numerical approximation of Feynman–Kac path integrals in the context of quantum statistical mechanics. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional Sobolev space by restricting the integrand to a subspace of all admissible paths. Through this process, a wide class of methods is derived, with each method corresponding to a different choice for the approximating subspace. It is shown that the traditional “short-time” approximation and “Fourier discretization” can be recovered by using linear and spectral basis functions, respectively. As an illustration of the flexibility afforded by the subspace approach, a novel method is formulated using cubic elements and is shown to have improved convergence properties when applied to model problems.