On computer simulation of Feynman-Kac path-integrals

Consider a path-integral Ex exp {∞toV(X(s))ds} f(X(t)) which is the solution to a diffusion version of the generalized Schrödingerʹs equation ∂u∂t = Hu, u(0,x) = ƒ(x). Here H = A + V, where A is an infinitesimal generator of a strong continuous Markov semigroup corresponding to the diffusion process {X(s), 0⩽s⩽t, X(0) = x}. To see a connection to quantum mechanics, take A = 12Δ and replace V by −V. Then one obtains H̄ = −H = −12Δ + V, which is a quantum mechanical Hamiltonian corresponding to a particle of mass 1 (in atomic units) subject to interaction with potential V. Path-integrals play a role in obtaining physical quantities such as ground state energies. This paper will be concerned with explanations of two approaches in the actual computer evaluations of path-integrals through simulations of the diffusion processes. The results will be presented by comparing, in concrete examples, the computational advantages or disadvantages depending on whether the diffusion process X(t) is ergodic or not.