en>Tobias Bergemann: Merge the very short "Properties" section into the lede. I think this helps to put Dini continuity into context. Feel free to revert if you prefer the previous version.

2012-06-11T17:42:25Z

Merge the very short "Properties" section into the lede. I think this helps to put Dini continuity into context. Feel free to revert if you prefer the previous version.

New page

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 [[conditional probability|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.<ref>Durrett, Iglehart: Functionals of Brownian Meander and Brownian Excursion, (1975)</ref>

==Definition==

A Brownian excursion process, <math>e</math>, is a [[Wiener process]] (or [[Brownian motion]]) [[conditional probability|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 <math>e</math> in terms of a Brownian motion process ''W'' (due to [[Paul Lévy (mathematician)|Paul Lévy]] and noted by [[Kiyoshi Itō]] and Henry P. McKean, Jr.<ref name=IM>Itô and McKean (1974, page 75)</ref>)
is in terms of the last time <math>\tau_{-} </math> that ''W'' hits zero before time 1 and the first time <math>\tau_{+} </math> that Brownian motion <math>W</math> hits zero after time 1:<ref name=IM/>

:<math>
\{ 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 \} .
</math>

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

:<math>
\{ 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 \} .
</math>

==Properties==

Vervaat's representation of a Brownian excursion has several consequences for various functions of <math>e</math>. In particular:

:<math>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) ,
</math>

(this can also be derived by explicit calculations<ref>[[Kai Lai Chung|Chung]] (1976)</ref><ref>Kennedy (1976) </ref>) and

:<math> \int_0^1 e(t) \, dt \ \stackrel{d}{=} \
\int_0^1 W_0 (t) \, dt - \inf_{0 \le t \le 1} W_0 (t) .
</math>

The following result holds:<ref name=DI>Durrett and Iglehart (1977)</ref>

:<math>E M_+ = \sqrt{\pi/2} \approx 1.25331 \ldots, \, </math>

and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:<ref name=DI/>

:<math>E M_+^2 \approx 1.64493 \ldots \ , \ \
Var(M_+) \approx 0.0741337 \ldots.</math>

Groeneboom (1989), Lemma 4.2 gives an expression for the [[Laplace transform]] of (the density) of
<math>\int_0^1 e(t) \, dt </math>. 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 [[Wiener process|Brownian motion]] <math>W</math> in terms of i.i.d Brownian excursions
and the least concave majorant (or greatest convex minorant) of <math>W</math>.

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

== Connections and applications ==

The Brownian excursion area

:<math>A_+ \equiv \int_0^1 e(t) \, dt </math>

arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.
,<ref>Wright, E. M. (1977). The number of connected sparsely edged graphs. J. Graph Th. 1, 317–330.</ref>
,<ref>Wright, E. M. (1980). The number connected sparsely edged graphs. III. Asymptotic results. J. Graph Th. 4, 393–407</ref>
,<ref>Spencer, J. (1997). Enumerating graphs and Brownian motion. Comm. Pure Appl. Math. 50, 291–294.</ref>
,<ref>[[Svante Janson|Janson, S.]] (2007). Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas.</ref>
,<ref>Flajolet, P. and Louchard, G. (2001). Analytic variations on the Airy distribution. Algorithmica 31, 361–377.</ref>
and the limit distribution of the Betti numbers of certain varieties in cohomology theory
.<ref>Reineke, M. (2005). Cohomology of noncommutative Hilbert schemes. Algebras and Representation Theory 8, 541–561.</ref>
Takacs (1991a) shows that <math>A_+</math> has density
:<math>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</math>
where <math>a_j </math> are the zeros of the Airy function and <math>U</math> is the [[confluent hypergeometric function]].
[[Svante Janson|Janson]] and Louchard (2007) show that

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

and

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

They also give higher-order expansions in both cases.

Janson (2007) gives moments of <math>A_+</math> and many other area functionals. In particular,

:<math>
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 \ .
</math>

Brownian excursions also arise in connection with
queuing problems,<ref>Iglehart, D. L. (1974). "Functional central limit theorms for random walks conditioned to stay positive." ''Ann. Probab.'', 2, 608–619.</ref>
railway traffic,<ref>Takacs, L. (1991a). A Bernoulli excursion and its various applications. Adv. in Appl. Probab. 23, 557–585.</ref><ref>Takacs, L. (1991b). "On a probability problem connected with railway traffic". ''J. Appl. Math. Stochastic Anal.'', 4, 263–292.</ref> and
the heights of random rooted binary trees.<ref>Takacs, L. (1994). "On the total heights of rooted binary trees". ''J. Combin. Theory Ser. B'', 61, 155–166.</ref>

== Related processes ==

* [[Brownian bridge]]
* [[Brownian meander]]
* reflected Brownian motion
* skew Brownian motion

==Notes==
<references />

== References ==
*{{cite journal
| author = [[Kai Lai Chung|Chung]], K. L.
| title = Maxima in Brownian excursions
| journal = Bull. Amer. Math Soc.
| volume = 81
| year = 1975
| pages = 742–745
| mr = 0373035
| url=http://projecteuclid.org/euclid.bams/1183537153
}}
*{{cite journal
| author = [[Kai Lai Chung|Chung]], K. L.
|title = Excursions in Brownian motion
|journal = Arkiv för Matematik
| volume = 14
| pages = 155–177
| year = 1976
| mr = 0467948
| url = http://link.springer.com/article/10.1007%2FBF02385832
}}
*{{cite journal
| author1 = Durrett, Richard T. |author2=Iglehart, Donald L.
| title =Functionals of Brownian meander and Brownian excursion
| journal= Annals of Probability
| volume=5
| pages=130–135
| year=1977
| mr=0436354
| url = http://projecteuclid.org/euclid.aop/1176995896 |jstor=2242808
}}
*{{cite journal
| first = Piet
|last = Groeneboom
|title=The concave majorant of Brownian motion
|journal= Annals of Probability
|volume=11
|pages=1016–1027
|year = 1983
|mr=714964
|url=http://projecteuclid.org/euclid.aop/1176993450 |jstor=2243513
}}
*{{cite journal
|first= Piet |last = Groeneboom
|title =Brownian motion with a parabolic drift and Airy functions
|journal= Probability Theory and Related Fields|volume = 81|pages=79–109
|year=1989| doi=10.1007/BF00343738|mr=981568
|url=http://link.springer.com/article/10.1007%2FBF00343738
}}
*{{cite book
| authorlink1=Kiyoshi Itō| last1=Itô|first1= Kiyosi
| authorlink2=Henry McKean| last2=McKean, Jr.| first2= Henry P.
| title = Diffusion Processes and their Sample Paths
| edition = Second printing, corrected
| publisher = Springer-Verlag, Berlin
| origyear = 1974
| year = 2013
| series = Classics in Mathematics
| mr = 0345224 |isbn=978-3540606291
}}
*{{cite journal
| author = Janson, Svante
| title = Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
| journal = Probability Surveys
| volume = 4
| pages = 80–145
| year = 2007
| mr = 2318402
}}
*{{cite journal
| author1 = Janson, Svante |author2=Louchard, Guy
| title = Tail estimates for the Brownian excursion area and other Brownian areas.
| journal = Electronic Journal of Probability
| volume = 12
| year = 2007
| pages = 1600–1632
| mr=2365879
| url=http://www.emis.de/journals/EJP-ECP/article/view/471.html
}}
*{{cite journal
| author= Kennedy, Douglas P.
| title = The distribution of the maximum Brownian excursion
| journal = J. Appl. Probability
| volume = 13
| year = 1976
| pages = 371–376
| mr=402955 A|jstor=3212843
}}
*{{cite book
| authorlink=Paul Lévy (mathematician)|last1=Lévy|first1= Paul
| title = Processus Stochastiques et Mouvement Brownien
| publisher = Gauthier-Villars, Paris
| year = 1948
| mr=0029120
}}
*{{cite journal
| author = Louchard, G.
| title = Kac's formula, Levy's local time and Brownian excursion
| journal = J. Appl. Probability
| volume = 21
| year = 1984
| pages = 479–499
| mr = 752014 | jstor = 3213611
}}
*{{cite journal
| author= Pitman, J. W.
| title = Remarks on the convex minorant of Brownian motion
| booktitle={Seminar on stochastic processes, 1982}
| series=Progr. Probab. Statist.
| volume = 5
|pages = 219–227
|publisher=Birkhauser, Boston
| year = 1983
| mr = 733673
}}
*{{cite book
| author1 = Revuz, Daniel |author2= Yor, Marc
| title = Continuous Martingales and Brownian Motion
| year = 2004
| series = Grundlehren der mathematischen Wissenschaften (Book 293) | publisher = Springer-Verlag, Berlin
| mr = 1725357
}}
*{{cite journal
| author = Vervaat, W.
| title = A relation between Brownian bridge and Brownian excursion
| journal = Annals of Probability
| volume = 7
| year = 1979
| pages = 143–149
| mr = 515820
|url=http://projecteuclid.org/euclid.aop/1176995155 |jstor=2242845
}}

{{Stochastic processes}}

{{DEFAULTSORT:Brownian Excursion}}
[[Category:Stochastic processes]]

Dini continuity - Revision history

en>Tobias Bergemann: Merge the very short "Properties" section into the lede. I think this helps to put Dini continuity into context. Feel free to revert if you prefer the previous version.