# Brownian excursion

A realization of Brownian Excursion.

In probability theory a Brownian excursion process is a stochastic process 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, ${\displaystyle e}$, 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 ${\displaystyle e}$ 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 ${\displaystyle \tau _{-}}$ that W hits zero before time 1 and the first time ${\displaystyle \tau _{+}}$ that Brownian motion ${\displaystyle W}$ hits zero after time 1:[2]

${\displaystyle \{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{{\frac {|W((1-t)\tau _{-}+t\tau _{+})|}{\sqrt {\tau _{+}-\tau _{-}}}}:\ 0\leq t\leq 1\right\}.}$

Let ${\displaystyle \tau _{m}}$ be the time that a Brownian bridge process ${\displaystyle W_{0}}$ achieves its minimum on [0, 1]. Vervaat (1979) shows that

${\displaystyle \{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{W_{0}(\tau _{m}+t{\bmod {1}})-W_{0}(\tau _{m}):\ 0\leq t\leq 1\right\}.}$

## Properties

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

${\displaystyle M_{+}\equiv \sup _{0\leq t\leq 1}e(t)\ {\stackrel {d}{=}}\ \sup _{0\leq t\leq 1}W_{0}(t)-\inf _{0\leq t\leq 1}W_{0}(t),}$

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

${\displaystyle \int _{0}^{1}e(t)\,dt\ {\stackrel {d}{=}}\ \int _{0}^{1}W_{0}(t)\,dt-\inf _{0\leq t\leq 1}W_{0}(t).}$

The following result holds:[5]

${\displaystyle EM_{+}={\sqrt {\pi /2}}\approx 1.25331\ldots ,\,}$

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

${\displaystyle EM_{+}^{2}\approx 1.64493\ldots \ ,\ \ \operatorname {Var} (M_{+})\approx 0.0741337\ldots .}$

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of ${\displaystyle \int _{0}^{1}e(t)\,dt}$. 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 ${\displaystyle W}$ in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of ${\displaystyle W}$.

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

${\displaystyle A_{+}\equiv \int _{0}^{1}e(t)\,dt}$

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 ${\displaystyle A_{+}}$ has density

${\displaystyle f_{A_{+}}(x)={\frac {2{\sqrt {6}}}{x^{2}}}\sum _{j=1}^{\infty }v_{j}^{2/3}e^{-v_{j}}U\left(-{\frac {5}{6}},{\frac {4}{3}};v_{j}\right)\ \ {\text{ with }}\ \ v_{j}={\frac {2|a_{j}|^{3}}{27x^{2}}}}$

where ${\displaystyle a_{j}}$ are the zeros of the Airy function and ${\displaystyle U}$ is the confluent hypergeometric function. Janson and Louchard (2007) show that

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

and

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

They also give higher-order expansions in both cases.

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

${\displaystyle E(A_{+})={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}},\ \ E(A_{+}^{2})={\frac {5}{12}}\approx 0.416666\ldots ,\ \ \operatorname {Var} (A_{+})={\frac {5}{12}}-{\frac {\pi }{8}}\approx .0239675\ldots \ .}$

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 (3): 291–294. doi:10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6.
9. ^ Janson, S. (2007). Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas.
10. ^ Flajolet P., Louchard G. (2001). "Analytic variations on the Airy distribution". Algorithmica. 31 (3): 361–377. CiteSeerX 10.1.1.27.3450. doi:10.1007/s00453-001-0056-0.
11. ^ Reineke M (2005). "Cohomology of noncommutative Hilbert schemes". Algebras and Representation Theory. 8 (4): 541–561. arXiv:math/0306185. doi:10.1007/s10468-005-8762-y.
12. ^ Iglehart D. L. (1974). "Functional central limit theorems for random walks conditioned to stay positive". Ann. Probab. 2 (4): 608–619. doi:10.1214/aop/1176996607.
13. ^ Takacs L (1991a). "A Bernoulli excursion and its various applications". Advances in Applied Probability. 23 (3): 557–585. doi:10.1017/s0001867800023739.
14. ^ Takacs L (1991b). "On a probability problem connected with railway traffic". J. Appl. Math. Stochastic Anal. 4: 263–292. doi:10.1155/S1048953391000011.
15. ^ Takacs L (1994). "On the total heights of rooted binary trees". J. Combin. Theory Ser. B. 61 (2): 155–166. doi:10.1006/jctb.1994.1041.