Stephen D. Bond, Brian B. Laird and Benedict J. Leimkuhler, On the approximation of Feynman-Kac path integrals, Journal of Computational Physics 185 (2003) 472-483.

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.

DOI: 10.1016/S0021-9991(02)00066-9

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@article{BLL2003,
  author  = {Stephen D. Bond and Brian B. Laird and Benedict J. Leimkuhler},
  title   = {On the approximation of Feynman-Kac path integrals},
  journal = {Journal of Computational Physics},
  volume  = 185,
  year	  = 2003,
  pages   = {472--483},
  doi	  = {10.1016/S0021-9991(02)00066-9}
}