Asymptotics of Some Feynman-Kac Functionals
Charles Hagwood
TL;DR
The paper studies the time average of Feynman-Kac-type functionals $e_K(t)$ for Markov processes and proves an explicit almost-sure limit: as $T\to\infty$, $\frac{1}{T}\int_0^T e_K(t) dt$ converges to $\int_I \frac{q(x)}{K(x)} \, m(dx)$, where $m$ is the invariant measure. The key technique combines a Cauchy Mean Value Theorem–style identity for $e_K(t)$ with a time-change argument and a decomposition of the outer integral into two parts, controlling each part in the limit. Under mild regularity assumptions on $q$ and $K$ (bounded $|q|$, $K$ bounded away from zero, and bounded derivative of $q/K$), the result links pathwise time averages to the stationary distribution via the action/reaction ratio $q/K$, with connections to parabolic PDEs and Schrödinger-type equations through the Feynman-Kac framework. The findings have implications for stochastic control, mathematical finance, and applied physics where long-term averages of occupation-type functionals are of interest.
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.