On Path Integrals for the High-dimensional Brownian Bridge

Let v be a bounded function with bounded support in Rd, d ≥ 3. Let x, y ∈ Rd. Let Z(t) denote the path integral of v along the path of a Brownian bridge in Rd which runs for time t, starting at x and ending at y. As t → ∞, 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.