On path integrals for the high-dimensional Brownian bridge

✍ Scribed by Robin Pemantle; Mathew D. Penrose

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 649 KB
• Volume: 44
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(92)90010-u

✦ Synopsis

Pemantle, R. and M.D. Penrose, On path integrals for the high-dimensional Brownian bridge, Journal of Computational and Applied Mathematics 44 (1992) 381-390. Let u be a bounded function with bounded support in [w d, d > 3. Let x, y E Rd. Let Z(t) denote the path integral of u along the path of a Brownian bridge in Rd which runs for time t, starting at x and ending at y.

As t -m, 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.