ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60J65, 60J60 Knight, Frank B., University of Illinois, U. S. A. On the moments of the area functional of Brownian bridge Let b_{t}; 0 ≤ t ≤ 1, be a Brownian bridge from 0 to 0, and let ℓ(u) be the total local time of |b_{t}| at u. For n ≥ 0, set G_{n}(x) = E((∫_{0}^{1} |b_{t}| dt)^{n} | ℓ(0) = x), i. e. the conditional moments of the absolute area given the local time at 0. We obtain a recursion formula for G_{n}(x), in the form

G_{n}(x) = (exp \frac{x^{2}}{16} ) ∑_{i=0}^{n−1} d_{n+2i,n}D_{−n−2i}(\frac{x}{2})

where D_{−k} are the parabolic cyclinder functions, d_{1,1} = \frac{1}{2} , and d_{k,n}, 1 < n, are determined by the recursion equations

d_{n+1+2i,n+1} = 2^{i−1}Γ(\frac{n + 1}{2} +i) ∑_{j=0}^{i∧(n−1)}(2^{j}Γ(\frac{n + 1}{2} +j)(n+1−j))^{−1}d_{n+2j,n} .