On computer simulation of Feynman-Kac path-integrals

• Published in: Journal of computational and applied mathematics Vol. 66; no. 1; pp. 333 - 336
• Main Author: Korzeniowski, Andrzej
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.1996; Elsevier
• Subjects: Classical and quantum physics: mechanics and fields; Diffusion random walks; Exact sciences and technology; Functional analytical methods; Global analysis, analysis on manifolds; Harmonic oscillator; Hydrogen atom; Mathematical methods in physics; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation and analysis; Numerical simulation; Numerical simulation, solution of equations; Physics; Quantum mechanics; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Hydrogen atom; 81S40; 65C05; 58D30; Diffusion random walks; Harmonic oscillator; Feynman integral; Feynman path integral; Partial differential equations; Random walk; Numerical simulation; Schroedinger equation; Harmonic oscillators
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/0377-0427(95)00170-0

Consider a path-integral E x exp {∞ t o V( X( s)) ds} f( X( t)) which is the solution to a diffusion version of the generalized Schrödinger's 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 = 1 2 Δ and replace V by − V. Then one obtains H ̄ = −H = − 1 2 Δ + 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.