Brownian excursion

File:Brownian bridge as a union of excursions.svg

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.Itô and McKean (1974, page 75))

is in terms of the last time $\backslash tau\_\{-\}$ that W hits zero before time 1 and the first time $\backslash tau\_\{+\}$ that Brownian motion $W$ hits zero after time 1:

:$$

\{ e(t) : \ {0 \le t \le 1} \} \ \stackrel{d}{=} \ \left \{ \frac

where the square root is due to the square root self-similarity of Wiener process. That is, $\backslash sqrt\; a\; W\_\{t/a\}$ is a Wiener process for any fixed constant $a\; >\; 0$.File:Vervaat's transformation.svg

Let $\backslash tau\_m$ 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 \bmod 1) - W_0 (\tau_m ): \ 0 \le t \le 1 \right \} .

This is sometimes called Vervaat's transformation.

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

:$M\_\{+\}\; \backslash equiv\; \backslash sup\_\{0\; \backslash le\; t\; \backslash le\; 1\}\; e(t)\; \backslash \; \backslash stackrel\{d\}\{=\}\; \backslash \; \backslash sup\_\{0\; \backslash le\; t\; \backslash le\; 1\}\; W\_0\; (t)\; -\; \backslash inf\_\{0\; \backslash le\; t\; \backslash le\; 1\}\; W\_0\; (t)\; ,$

(this can also be derived by explicit calculationsChung (1976)Kennedy (1976)) and

:$\backslash int\_0^1\; e(t)\; \backslash ,\; dt\; \backslash \; \backslash stackrel\{d\}\{=\}\; \backslash$

\int_0^1 W_0 (t) \, dt - \inf_{0 \le t \le 1} W_0 (t) .

The following result holds:Durrett and Iglehart (1977)

:$E\; M\_+\; =\; \backslash sqrt\{\backslash pi/2\}\; \backslash approx\; 1.25331\; \backslash ldots,\; \backslash ,$

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

:$E\; M\_+^2\; \backslash approx\; 1.64493\; \backslash ldots\; \backslash \; ,\; \backslash \; \backslash$

\operatorname{Var}(M_+) \approx 0.0741337 \ldots.

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of $\backslash int\_0^1\; e(t)\; \backslash ,\; dt$. 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.

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

:$A\_+\; \backslash equiv\; \backslash int\_0^1\; e(t)\; \backslash ,\; dt$

arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.{{Cite journal |doi = 10.1002/jgt.3190010407|title = The number of connected sparsely edged graphs|journal = Journal of Graph Theory|volume = 1|issue = 4|pages = 317–330|year = 1977|last1 = Wright|first1 = E. M.|authorlink1=E. M. Wright}}{{Cite journal |doi = 10.1002/jgt.3190040409|title = The number of connected sparsely edged graphs. III. Asymptotic results|journal = Journal of Graph Theory|volume = 4|issue = 4|pages = 393–407|year = 1980|last1 = Wright|first1 = E. M.|authorlink1=E. M. Wright}}{{cite journal | author = Spencer J | year = 1997 | title = Enumerating graphs and Brownian motion | journal = Communications on Pure and Applied Mathematics| volume = 50 | issue = 3| pages = 291–294 | doi=10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6}}{{Cite journal | doi=10.1214/07-PS104| title=Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas| journal=Probability Surveys| volume=4| pages=80–145| year=2007| last1=Janson| first1=Svante| bibcode=2007arXiv0704.2289J| arxiv=0704.2289| s2cid=14563292}}{{cite journal | author = Flajolet, P. |author2=Louchard, G. | year = 2001 | title = Analytic variations on the Airy distribution | journal = Algorithmica | volume = 31 | issue = 3| pages = 361–377 | doi=10.1007/s00453-001-0056-0| citeseerx = 10.1.1.27.3450 |s2cid=6522038 }} and the limit distribution of the Betti numbers of certain varieties in cohomology theory.{{cite journal | author = Reineke M | year = 2005 | title = Cohomology of noncommutative Hilbert schemes | journal = Algebras and Representation Theory | volume = 8 | issue = 4| pages = 541–561 | doi=10.1007/s10468-005-8762-y| arxiv = math/0306185| s2cid = 116587916 }} Takacs (1991a) shows that $A\_+$ has density

:$f\_\{A\_+\}\; (x)\; =\; \backslash frac\{2\; \backslash sqrt\{6\}\}\{x^2\}\; \backslash sum\_\{j=1\}^\backslash infty\; v\_j^\{2/3\}\; e^\{-v\_j\}\; U\backslash left\; (\; -\; \backslash frac\{5\}\{6\}\; ,\; \backslash frac\{4\}\{3\};\; v\_j\; \backslash right\; )\; \backslash \; \backslash \; \backslash text\{\; with\; \}\; \backslash \; \backslash \; v\_j\; =\; \backslash frac\{2\; |a\_j|^3\}\{27x^2\}$

where $a\_j$ are the zeros of the Airy function and $U$ is the confluent hypergeometric function.

Janson and Louchard (2007) show that

:$f\_\{A\_+\}\; (x)\; \backslash sim\; \backslash frac\{72\; \backslash sqrt\{6\}\}\{\backslash sqrt\{\backslash pi\}\}\; x^2\; e^\{-\; 6\; x^2\}\; \backslash \; \backslash \; \backslash text\{\; as\; \}\; \backslash \; \backslash \; x\; \backslash rightarrow\; \backslash infty,$

and

:$P(A\_+\; >\; x)\; \backslash sim\; \backslash frac\{6\; \backslash sqrt\{6\}\}\{\backslash sqrt\{\backslash pi\}\}\; x\; e^\{-\; 6x^2\}\; \backslash \; \backslash \; \backslash text\{\; as\; \}\; \backslash \; \backslash \; x\; \backslash rightarrow\; \backslash infty.$

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,{{cite journal | author = Iglehart D. L. | year = 1974 | title = Functional central limit theorems for random walks conditioned to stay positive | journal = The Annals of Probability| volume = 2 | issue = 4| pages = 608–619 | doi=10.1214/aop/1176996607| doi-access = free }}

railway traffic,{{cite journal | author = Takacs L | year = 1991a | title = A Bernoulli excursion and its various applications | journal = Advances in Applied Probability | volume = 23 | issue = 3| pages = 557–585 | doi=10.1017/s0001867800023739}}{{cite journal | author = Takacs L | year = 1991b | title = On a probability problem connected with railway traffic | journal = Journal of Applied Mathematics and Stochastic Analysis| volume = 4 | pages = 263–292 | doi = 10.1155/S1048953391000011 | doi-access = free }} and the heights of random rooted binary trees.{{cite journal | author = Takacs L | year = 1994 | title = On the Total Heights of Random Rooted Binary Trees| journal = Journal of Combinatorial Theory, Series B| volume = 61 | issue = 2| pages = 155–166 | doi=10.1006/jctb.1994.1041| doi-access = free}}

• Brownian bridge
• Brownian meander
• reflected Brownian motion
• skew Brownian motion

• {{cite journal

|author=Chung, K. L.

|author-link=Kai Lai Chung

|title=Maxima in Brownian excursions

|journal=Bulletin of the American Mathematical Society

|volume=81

|issue=4

|year=1975

|pages=742–745

|mr=0373035

|url=http://projecteuclid.org/euclid.bams/1183537153

|doi=10.1090/s0002-9904-1975-13852-3

|doi-access=free

}}

• {{cite journal

|author=Chung, K. L.

|title=Excursions in Brownian motion

|journal=Arkiv för Matematik

|volume=14

|issue=1

|pages=155–177

|year=1976

|mr=0467948

|doi=10.1007/bf02385832

|bibcode=1976ArM....14..155C

|doi-access=free

}}

• {{cite journal

|author1=Durrett, Richard T. |author2=Iglehart, Donald L.

|title=Functionals of Brownian meander and Brownian excursion

|journal=Annals of Probability

|volume=5

|issue=1

|pages=130–135

|year=1977

|mr=0436354

|jstor=2242808

|doi=10.1214/aop/1176995896

|doi-access=free

}}

• {{cite journal

|first=Piet

|last=Groeneboom

|title=The concave majorant of Brownian motion

|journal=Annals of Probability

|volume=11

|issue=4

|pages=1016–1027

|year=1983

|mr=714964

|jstor=2243513

|doi=10.1214/aop/1176993450

|doi-access=free

}}

• {{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

|s2cid=119980629

|url=https://ir.cwi.nl/pub/6435

|doi-access=free}}

• {{cite book

|author-link1=Kiyoshi Itō |last1=Itô |first1=Kiyosi

|author-link2=Henry McKean |last2=McKean, Jr. |first2=Henry P.

|title=Diffusion Processes and their Sample Paths

|edition=Second printing, corrected

|publisher=Springer-Verlag, Berlin

|orig-year=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

|doi=10.1214/07-ps104

|arxiv=0704.2289

|bibcode=2007arXiv0704.2289J

|s2cid=14563292

}}

• {{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

|doi=10.1214/ejp.v12-471

|arxiv=0707.0991

|bibcode=2007arXiv0707.0991J

|s2cid=6281609

}}

• {{cite journal

|author=Kennedy, Douglas P.

|title=The distribution of the maximum Brownian excursion

|journal=Journal of Applied Probability

|volume=13

|issue=2

|year=1976

|pages=371–376

|mr=0402955

|jstor=3212843

|doi=10.2307/3212843

|s2cid=222386970

}}

• {{cite book

|author-link=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=Journal of Applied Probability

|volume=21

|issue=3

|year=1984

|pages=479–499

|mr=752014 |jstor=3213611

|doi=10.2307/3213611

|s2cid=123640749

}}

• {{cite conference

|author=Pitman, J. W.

|title=Remarks on the convex minorant of Brownian motion

|book-title=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

|volume=293

|year=2004

|series=Grundlehren der mathematischen Wissenschaften

|publisher=Springer-Verlag, Berlin

|mr=1725357

|doi=10.1007/978-3-662-06400-9

|isbn=978-3-642-08400-3

}}

• {{cite journal

|author=Vervaat, W.

|title=A relation between Brownian bridge and Brownian excursion

|journal=Annals of Probability

|volume=7

|issue=1

|year=1979

|pages=143–149

|mr=515820

|jstor=2242845

|doi=10.1214/aop/1176995155

|doi-access=free

}}

{{Stochastic processes}}

{{DEFAULTSORT:Brownian Excursion}}

Category:Wiener process