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On Path Integrals for the High-dimensional Brownian Bridge

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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).