Design of high-order short-time approximations as a problem of matching the covariance of a Brownian motion

TL;DR

This paper addresses the challenge of designing high-order short-time approximations for the Feynman-Kac formula by reformulating the problem as matching the covariance of a Brownian motion, enabling stable and potential-only based discretizations.

Contribution

It demonstrates how to construct covariance matrices for Gaussian processes that match generalized moments of Brownian motion, facilitating high-order approximations.

Findings

Covariance-based construction of Gaussian processes for approximation.

Stable high-order short-time approximation methods.

Potential-only knowledge suffices for accurate discretization.

Abstract

One of the outstanding problems in the numerical discretization of the Feynman-Kac formula calls for the design of arbitrary-order short-time approximations that are constructed in a stable way, yet only require knowledge of the potential function. In essence, the problem asks for the development of a functional analogue to the Gauss quadrature technique for one-dimensional functions. In PRE 69, 056701 (2004), it has been argued that the problem of designing an approximation of order \nu is equivalent to the problem of constructing discrete-time Gaussian processes that are supported on finite-dimensional probability spaces and match certain generalized moments of the Brownian motion. Since Gaussian processes are uniquely determined by their covariance matrix, it is tempting to reformulate the moment-matching problem in terms of the covariance matrix alone. Here, we show how this can be accomplished.