# Brownian excursion

In probability theory a Brownian excursion process is a stochastic processes that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems.[1]

## Definition

A Brownian excursion process, $e$, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive.

Another representation of a Brownian excursion $e$ in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyoshi Itō and Henry P. McKean, Jr.[2]) is in terms of the last time $\tau_{-}$ that W hits zero before time 1 and the first time $\tau_{+}$ that Brownian motion $W$ hits zero after time 1:[2]

$\{ e(t) : \ {0 \le t \le 1} \} \ \stackrel{d}{=} \ \left \{ \frac{|W((1-t) \tau_{-} + t \tau_{+} )|}{\sqrt{\tau_+ - \tau_{-}}} : \ 0 \le t \le 1 \right \} .$

Let $\tau_m$ be the time that a Brownian bridge process $W_0$ achieves its minimum on [0, 1]. Vervaat (1979) shows that

$\{ e(t) : \ {0\le t \le 1} \} \ \stackrel{d}{=} \ \left \{ W_0 ( \tau_m + t \text{ mod } 1) - W_0 (\tau_m ): \ 0 \le t \le 1 \right \} .$

## Properties

Vervaat's representation of a Brownian excursion has several consequences for various functions of $e$. In particular:

$M_{+} \equiv \sup_{0 \le t \le 1} e(t) \ \stackrel{d}{=} \ \sup_{0 \le t \le 1} W_0 (t) - \inf_{0 \le t \le 1} W_0 (t) ,$

(this can also be derived by explicit calculations[3][4]) and

$\int_0^1 e(t) \, dt \ \stackrel{d}{=} \ \int_0^1 W_0 (t) \, dt - \inf_{0 \le t \le 1} W_0 (t) .$

The following result holds:[5]

$E M_+ = \sqrt{\pi/2} \approx 1.25331 \ldots, \,$

and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:[5]

$E M_+^2 \approx 1.64493 \ldots \ , \ \ Var(M_+) \approx 0.0741337 \ldots.$

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of $\int_0^1 e(t) \, dt$. A formula for a certain double transform of the distribution of this area integral is given by Louchard (1984).

Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion $W$ in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of $W$.

For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII.

## Connections and applications

The Brownian excursion area

$A_+ \equiv \int_0^1 e(t) \, dt$

arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g. ,[6] ,[7] ,[8] ,[9] ,[10] and the limit distribution of the Betti numbers of certain varieties in cohomology theory .[11] Takacs (1991a) shows that $A_+$ has density

$f_{A_+} (x) = \frac{2 \sqrt{6}}{x^2} \sum_{j=1}^\infty v_j^{2/3} e^{-v_j} U\left ( - \frac{5}{6} , \frac{4}{3}; v_j \right ) \ \ \mbox{with} \ \ v_j = 2 |a_j|^3 / 27x^2$

where $a_j$ are the zeros of the Airy function and $U$ is the confluent hypergeometric function. Janson and Louchard (2007) show that

$f_{A_+} (x) \sim \frac{72 \sqrt{6}}{\sqrt{\pi}} x^2 e^{- 6 x^2} \ \ \mbox{as} \ \ x \rightarrow \infty,$

and

$P(A_+ > x) \sim \frac{6 \sqrt{6}}{\sqrt{\pi}} x e^{- 6x^2} \ \ \mbox{as} \ \ x \rightarrow \infty.$

They also give higher-order expansions in both cases.

Janson (2007) gives moments of $A_+$ and many other area functionals. In particular,

$E (A_+) = \frac{1}{2} \sqrt{\frac{\pi}{2}}, \ \ E(A_+^2) = \frac{5}{12} \approx .416666 \ldots, \ \ Var(A_+) = \frac{5}{12} - \frac{\pi}{8} \approx .0239675 \ldots \ .$

Brownian excursions also arise in connection with queuing problems,[12] railway traffic,[13][14] and the heights of random rooted binary trees.[15]

## Notes

1. ^ Durrett, Iglehart: Functionals of Brownian Meander and Brownian Excursion, (1975)
2. ^ a b Itô and McKean (1974, page 75)
3. ^ Chung (1976)
4. ^ Kennedy (1976)
5. ^ a b Durrett and Iglehart (1977)
6. ^ Wright, E. M. (1977). The number of connected sparsely edged graphs. J. Graph Th. 1, 317–330.
7. ^ Wright, E. M. (1980). The number connected sparsely edged graphs. III. Asymptotic results. J. Graph Th. 4, 393–407
8. ^ Spencer, J. (1997). Enumerating graphs and Brownian motion. Comm. Pure Appl. Math. 50, 291–294.
9. ^ Janson, S. (2007). Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas.
10. ^ Flajolet, P. and Louchard, G. (2001). Analytic variations on the Airy distribution. Algorithmica 31, 361–377.
11. ^ Reineke, M. (2005). Cohomology of noncommutative Hilbert schemes. Algebras and Representation Theory 8, 541–561.
12. ^ Iglehart, D. L. (1974). "Functional central limit theorems for random walks conditioned to stay positive." Ann. Probab., 2, 608–619.
13. ^ Takacs, L. (1991a). A Bernoulli excursion and its various applications. Adv. in Appl. Probab. 23, 557–585.
14. ^ Takacs, L. (1991b). "On a probability problem connected with railway traffic". J. Appl. Math. Stochastic Anal., 4, 263–292.
15. ^ Takacs, L. (1994). "On the total heights of rooted binary trees". J. Combin. Theory Ser. B, 61, 155–166.