The Annals of Applied Probability

On the distribution of Brownian areas

Mihael Perman and Jon A. Wellner

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We study the distributions of the areas under the positive parts of a Brownian motion process B and a Brownian bridge process U: with $A^+ = \int_0^1 B^+ (t) dt$ and $A_0^+ = \int_0^1 U^+ (t) dt$, we use excursion theory to show that the Laplace transforms $\Psi^+ (s) = E \exp (-sA^+)$ and $\Psi_0^+ (s) = E \exp (-sA_0^+)$ of $A^+$ of $A_0^+$ satisfy $$\int_0^{\infty} e^{-\lambda s \Psi +} (\sqrt{2} s^{3/2}) ds = \frac{\lambda^{-1/2} Ai(\lambda) + (1/3 - \int_0^{\lambda} Ai(t) dt)}{\sqrt{\lambda} Ai(\lambda) - Ai (\lambda)}.$$ and $$\int_0^{\infty} \frac{e^{-\lambda s}{\sqrt{s}} \Psi_0^+ (\sqrt{2} s^{3/2}) ds = 2 \sqrt{\pi} \frac{Ai(\lambda)}{\sqrt{\lambda} Ai'(\lambda) - Ai(\lambda)}.$$ where Ai is Airy's function. At the same time, our approach via excursion theory unifies previous calculations of this type due to Kac, Groeneboom, Louchard, Shepp and Takács for other Brownian areas. Similarly, we use excursion theory to obtain recursion formulas for the moments of the "positive part" areas. We have not yet succeeded in inverting the double Laplace transforms because of the structure of the function appearing in the denominators, namely, $\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)$.

Article information

Ann. Appl. Probab., Volume 6, Number 4 (1996), 1091-1111.

First available in Project Euclid: 24 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G99: None of the above, but in this section
Secondary: 60E05: Distributions: general theory

Absolute value Airy functions area asymptotic distribution Brownian bridge Brownian excursion Brownian motion Feynman-Kac inversion moments positive part recursion formulas


Perman, Mihael; Wellner, Jon A. On the distribution of Brownian areas. Ann. Appl. Probab. 6 (1996), no. 4, 1091--1111. doi:10.1214/aoap/1035463325.

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