Asymptotics of Some Feynman-Kac Functionals

TL;DR

This paper investigates the long-term behavior of time averages of Feynman-Kac functionals associated with Markov processes, providing asymptotic expressions useful in various scientific and engineering fields.

Contribution

It derives an explicit asymptotic expression for the time average of Feynman-Kac functionals, extending understanding of their long-term properties.

Findings

Asymptotic expression for the time average derived

Long-term averages serve as better predictors in real-time applications

Results applicable to physics, finance, control theory, and PDEs

Abstract

Methods were initiated by Mark Kac and Richard Feynman to evaluate random functionals of the form $\int^t_0V(X_s)ds$ for a nonnegative $V$ and a Markov process $X_t$. Their results evolved into the well known Feynman Kac formula. Functionals of this type appear in both theoretical and applied applications in partial differential equations, quantum physics, mathematical finance, control theory, etc. Here the time average of one such functional associated with the Feynman Kac formula is studied. In real time applications where only the path is observed, the time average often is a better predictor than the functional at its last observation point.. It represents quantities, e.g., the long term average cost or wealth, the long term average velocity. As a statistic, it is of interest to determine if it has an asymptotic limit and to determine that limit. An expression is derived for its asymptotic time average.