# Brownian excursion

In probability theory a Brownian excursion process (BPE) 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.

## Constructions

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. Although note that, since the probability for an unrestricted Brownian bridge to be positive is zero, the conditioning requires care.
Another representation of a Brownian excursion $e$ in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyosi Itô and Henry P. McKean, Jr.) 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:
    $\{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{{\frac {|W((1-t)\tau _{-}+t\tau _{+})|}{\sqrt {\tau _{+}-\tau _{-}}}}:\ 0\leq t\leq 1\right\}.$
where the square root is due to the square root self-similarity of Wiener process. That is, ${\sqrt {a}}W_{t/a}$ is a Wiener process for any fixed constant $a>0$.
Let $\tau _{m}$ be the time that a Brownian bridge process $W_{0}$ achieves its minimum on [0, 1]. Vervaat (1979) shows that
    $\{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{W_{0}(\tau _{m}+t{\bmod {1}})-W_{0}(\tau _{m}):\ 0\leq t\leq 1\right\}.$
This is sometimes called Vervaat's transformation.

## Properties

Vervaat's representation of a Brownian excursion has several consequences for various functions of $e$. In particular:
    $M_{+}\equiv \sup _{0\leq t\leq 1}e(t)\ {\stackrel {d}{=}}\ \sup _{0\leq t\leq 1}W_{0}(t)-\inf _{0\leq t\leq 1}W_{0}(t),$
(this can also be derived by explicit calculations) and
    $\int _{0}^{1}e(t)\,dt\ {\stackrel {d}{=}}\ \int _{0}^{1}W_{0}(t)\,dt-\inf _{0\leq t\leq 1}W_{0}(t).$
The following result holds:
    $EM_{+}={\sqrt {\pi /2}}\approx 1.25331\ldots ,\,$
and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:
    $EM_{+}^{2}\approx 1.64493\ldots \ ,\ \ \operatorname {Var} (M_{+})\approx 0.0741337\ldots .$
Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of $\int _{0}^{1}e(t)\,dt$. 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

With probability 1, a Wiener process is continuous, which means the set on which it is non-zero is an open subset of the real line, thus it is the union of countably many Brownian excursions.
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. and the limit distribution of the Betti numbers of certain varieties in cohomology theory. Takacs (1991a) shows that $A_{+}$ has density
    $f_{A_{+}}(x)={\frac {2{\sqrt {6}}}{x^{2}}}\sum _{j=1}^{\infty }v_{j}^{2/3}e^{-v_{j}}U\left(-{\frac {5}{6}},{\frac {4}{3}};v_{j}\right)\ \ {\text{ with }}\ \ v_{j}={\frac {2|a_{j}|^{3}}{27x^{2}}}$
where $a_{j}$ are the zeros of the Airy function and $U$ is the confluent hypergeometric function. Janson and Louchard (2007) show that
    $f_{A_{+}}(x)\sim {\frac {72{\sqrt {6}}}{\sqrt {\pi }}}x^{2}e^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty ,$
and
    $P(A_{+}>x)\sim {\frac {6{\sqrt {6}}}{\sqrt {\pi }}}xe^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty .$
They also give higher-order expansions in both cases.
Janson (2007) gives moments of $A_{+}$ and many other area functionals. In particular,
    $E(A_{+})={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}},\ \ E(A_{+}^{2})={\frac {5}{12}}\approx 0.416666\ldots ,\ \ \operatorname {Var} (A_{+})={\frac {5}{12}}-{\frac {\pi }{8}}\approx .0239675\ldots \ .$
Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees.

## Related processes

* Brownian bridge
* Brownian meander
* reflected Brownian motion
* skew Brownian motion