On Path Integrals for the High-dimensional Brownian Bridge

By Robin Pemantle and Mathew D. Penrose.

Let $v$ be a bounded function with bounded support in $d$-dimensional space $R^d$, with $d >2$. Let $x,y$ be elements of $R^d$. Let $Z(t)$ denote the path integral of $v$ along the path of a Brownian bridge in $R^d$ which runs for time $t$, starting at $x$ and ending at $y$. As $t$ goes to infinity, it is perhaps evident that the distribution of $Z(t)$ converges weakly to that of the sum of the integrals of $v$ along the paths of two independent Brownian motions, starting at $x$ and $y$ and running forever. Here we prove a stronger result, namely convergence of the corresponding moment generating functions and of moments. This result is needed for applications in physics.

Journal of Computational and Applied Mathematics 44, 381-390 (1992).