Brownian excursion

Stochastic process

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

[edit]

A Brownian excursion process,  e {\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  e {\displaystyle e} in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyosi Itô and Henry P. McKean, Jr.[2]) is in terms of the last time  τ − {\displaystyle \tau _{-}} that W hits zero before time 1 and the first time  τ + {\displaystyle \tau _{+}} that Brownian motion  W {\displaystyle W} hits zero after time 1:[2]

{ e ( t ) : 0 ≤ t ≤ 1 } = d { | W ( ( 1 − t ) τ − + t τ + ) | τ + − τ − : 0 ≤ t ≤ 1 } . {\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  τ m {\displaystyle \tau _{m}} be the time that a Brownian bridge process  W 0 {\displaystyle W_{0}} achieves its minimum on [0, 1]. Vervaat (1979) shows that

{ e ( t ) : 0 ≤ t ≤ 1 } = d { W 0 ( τ m + t mod 1 ) − W 0 ( τ m ) : 0 ≤ t ≤ 1 } . {\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

[edit]

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

M + ≡ sup 0 ≤ t ≤ 1 e ( t ) = d sup 0 ≤ t ≤ 1 W 0 ( t ) − inf 0 ≤ t ≤ 1 W 0 ( t ) , {\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

∫ 0 1 e ( t ) d t = d ∫ 0 1 W 0 ( t ) d t − inf 0 ≤ t ≤ 1 W 0 ( t ) . {\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]

E M + = π / 2 ≈ 1.25331 … , {\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]

E M + 2 ≈ 1.64493 … , Var ⁡ ( M + ) ≈ 0.0741337 … . {\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  ∫ 0 1 e ( t ) d t {\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 W {\displaystyle W} in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of  W {\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

[edit]

The Brownian excursion area

A + ≡ ∫ 0 1 e ( t ) d t {\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  A + {\displaystyle A_{+}} has density

f A + ( x ) = 2 6 x 2 ∑ j = 1 ∞ v j 2 / 3 e − v j U ( − 5 6 , 4 3 ; v j ) with v j = 2 | a j | 3 27 x 2 {\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  a j {\displaystyle a_{j}} are the zeros of the Airy function and  U {\displaystyle U} is the confluent hypergeometric function. Janson and Louchard (2007) show that

f A + ( x ) ∼ 72 6 π x 2 e − 6 x 2 as x -> ∞ , {\displaystyle f_{A_{+}}(x)\sim {\frac {72{\sqrt {6}}}{\sqrt {\pi }}}x^{2}e^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty ,}

and

P ( A + > x ) ∼ 6 6 π x e − 6 x 2 as x -> ∞ . {\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  A + {\displaystyle A_{+}} and many other area functionals. In particular,

E ( A + ) = 1 2 π 2 , E ( A + 2 ) = 5 12 ≈ 0.416666 … , Var ⁡ ( A + ) = 5 12 − π 8 ≈ .0239675 … . {\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]

## Related processes

[edit]

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

## Notes

[edit]

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". Journal of Graph Theory. 1 (4): 317–330\. doi:10.1002/jgt.3190010407.317-330&rft.date=1977&rft_id=info:doi/10.1002/jgt.3190010407&rft.aulast=Wright&rft.aufirst=E.+M.&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
7. ^ Wright, E. M. (1980). "The number of connected sparsely edged graphs. III. Asymptotic results". Journal of Graph Theory. 4 (4): 393–407\. doi:10.1002/jgt.3190040409.393-407&rft.date=1980&rft_id=info:doi/10.1002/jgt.3190040409&rft.aulast=Wright&rft.aufirst=E.+M.&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
8. ^ Spencer J (1997). "Enumerating graphs and Brownian motion". Communications on Pure and Applied Mathematics. 50 (3): 291–294\. doi:10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6.291-294&rft.date=1997&rft_id=info:doi/10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6&rft.au=Spencer+J&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
9. ^ Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys. 4: 80–145\. arXiv:0704.2289. Bibcode:2007arXiv0704.2289J. doi:10.1214/07-PS104. S2CID 14563292.80-145&rft.date=2007&rft_id=info:arxiv/0704.2289&rft_id=https://api.semanticscholar.org/CorpusID:14563292#id-name=S2CID&rft_id=info:doi/10.1214/07-PS104&rft_id=info:bibcode/2007arXiv0704.2289J&rft.aulast=Janson&rft.aufirst=Svante&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
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. S2CID 6522038.361-377&rft.date=2001&rft_id=https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.3450#id-name=CiteSeerX&rft_id=https://api.semanticscholar.org/CorpusID:6522038#id-name=S2CID&rft_id=info:doi/10.1007/s00453-001-0056-0&rft.au=Flajolet,+P.&rft.au=Louchard,+G.&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
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. S2CID 116587916.541-561&rft.date=2005&rft_id=info:arxiv/math/0306185&rft_id=https://api.semanticscholar.org/CorpusID:116587916#id-name=S2CID&rft_id=info:doi/10.1007/s10468-005-8762-y&rft.au=Reineke+M&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
12. ^ Iglehart D. L. (1974). "Functional central limit theorems for random walks conditioned to stay positive". The Annals of Probability. 2 (4): 608–619\. doi:10.1214/aop/1176996607.608-619&rft.date=1974&rft_id=info:doi/10.1214/aop/1176996607&rft.au=Iglehart+D.+L.&rft_id=https://doi.org/10.1214%2Faop%2F1176996607&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
13. ^ Takacs L (1991a). "A Bernoulli excursion and its various applications". Advances in Applied Probability. 23 (3): 557–585\. doi:10.1017/s0001867800023739.557-585&rft.date=1991&rft_id=info:doi/10.1017/s0001867800023739&rft.au=Takacs+L&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
14. ^ Takacs L (1991b). "On a probability problem connected with railway traffic". Journal of Applied Mathematics and Stochastic Analysis. 4: 263–292\. doi:10.1155/S1048953391000011.263-292&rft.date=1991&rft_id=info:doi/10.1155/S1048953391000011&rft.au=Takacs+L&rft_id=https://doi.org/10.1155%2FS1048953391000011&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
15. ^ Takacs L (1994). "On the Total Heights of Random Rooted Binary Trees". Journal of Combinatorial Theory, Series B. 61 (2): 155–166\. doi:10.1006/jctb.1994.1041.155-166&rft.date=1994&rft_id=info:doi/10.1006/jctb.1994.1041&rft.au=Takacs+L&rft_id=https://doi.org/10.1006%2Fjctb.1994.1041&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">

## References

[edit]

* Chung, K. L. (1975). "Maxima in Brownian excursions". Bulletin of the American Mathematical Society. 81 (4): 742–745\. doi:10.1090/s0002-9904-1975-13852-3. MR 0373035.742-745&rft.date=1975&rft_id=info:doi/10.1090/s0002-9904-1975-13852-3&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0373035#id-name=MR&rft.au=Chung,+K.+L.&rft_id=http://projecteuclid.org/euclid.bams/1183537153&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Chung, K. L. (1976). "Excursions in Brownian motion". Arkiv för Matematik. 14 (1): 155–177\. Bibcode:1976ArM....14..155C. doi:10.1007/bf02385832. MR 0467948.155-177&rft.date=1976&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0467948#id-name=MR&rft_id=info:doi/10.1007/bf02385832&rft_id=info:bibcode/1976ArM....14..155C&rft.au=Chung,+K.+L.&rft_id=https://doi.org/10.1007%2Fbf02385832&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Durrett, Richard T.; Iglehart, Donald L. (1977). "Functionals of Brownian meander and Brownian excursion". Annals of Probability. 5 (1): 130–135\. doi:10.1214/aop/1176995896. JSTOR 2242808. MR 0436354.130-135&rft.date=1977&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0436354#id-name=MR&rft_id=https://www.jstor.org/stable/2242808#id-name=JSTOR&rft_id=info:doi/10.1214/aop/1176995896&rft.au=Durrett,+Richard+T.&rft.au=Iglehart,+Donald+L.&rft_id=https://doi.org/10.1214%2Faop%2F1176995896&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Groeneboom, Piet (1983). "The concave majorant of Brownian motion". Annals of Probability. 11 (4): 1016–1027\. doi:10.1214/aop/1176993450. JSTOR 2243513. MR 0714964.1016-1027&rft.date=1983&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=714964#id-name=MR&rft_id=https://www.jstor.org/stable/2243513#id-name=JSTOR&rft_id=info:doi/10.1214/aop/1176993450&rft.aulast=Groeneboom&rft.aufirst=Piet&rft_id=https://doi.org/10.1214%2Faop%2F1176993450&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields. 81: 79–109\. doi:10.1007/BF00343738. MR 0981568. S2CID 119980629.79-109&rft.date=1989&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=981568#id-name=MR&rft_id=https://api.semanticscholar.org/CorpusID:119980629#id-name=S2CID&rft_id=info:doi/10.1007/BF00343738&rft.aulast=Groeneboom&rft.aufirst=Piet&rft_id=https://ir.cwi.nl/pub/6435&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Itô, Kiyosi; McKean, Jr., Henry P. (2013) [1974]. Diffusion Processes and their Sample Paths. Classics in Mathematics (Second printing, corrected ed.). Springer-Verlag, Berlin. ISBN 978-3540606291. MR 0345224.
* Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys. 4: 80–145\. arXiv:0704.2289. Bibcode:2007arXiv0704.2289J. doi:10.1214/07-ps104. MR 2318402. S2CID 14563292.80-145&rft.date=2007&rft_id=https://api.semanticscholar.org/CorpusID:14563292#id-name=S2CID&rft_id=info:bibcode/2007arXiv0704.2289J&rft_id=info:arxiv/0704.2289&rft_id=info:doi/10.1214/07-ps104&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=2318402#id-name=MR&rft.au=Janson,+Svante&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Janson, Svante; Louchard, Guy (2007). "Tail estimates for the Brownian excursion area and other Brownian areas". Electronic Journal of Probability. 12: 1600–1632\. arXiv:0707.0991. Bibcode:2007arXiv0707.0991J. doi:10.1214/ejp.v12-471. MR 2365879. S2CID 6281609.1600-1632&rft.date=2007&rft_id=https://api.semanticscholar.org/CorpusID:6281609#id-name=S2CID&rft_id=info:bibcode/2007arXiv0707.0991J&rft_id=info:arxiv/0707.0991&rft_id=info:doi/10.1214/ejp.v12-471&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=2365879#id-name=MR&rft.au=Janson,+Svante&rft.au=Louchard,+Guy&rft_id=http://www.emis.de/journals/EJP-ECP/article/view/471.html&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Kennedy, Douglas P. (1976). "The distribution of the maximum Brownian excursion". Journal of Applied Probability. 13 (2): 371–376\. doi:10.2307/3212843. JSTOR 3212843. MR 0402955. S2CID 222386970.371-376&rft.date=1976&rft_id=https://api.semanticscholar.org/CorpusID:222386970#id-name=S2CID&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0402955#id-name=MR&rft_id=https://www.jstor.org/stable/3212843#id-name=JSTOR&rft_id=info:doi/10.2307/3212843&rft.au=Kennedy,+Douglas+P.&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Lévy, Paul (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. MR 0029120.
* Louchard, G. (1984). "Kac's formula, Levy's local time and Brownian excursion". Journal of Applied Probability. 21 (3): 479–499\. doi:10.2307/3213611. JSTOR 3213611. MR 0752014. S2CID 123640749.479-499&rft.date=1984&rft_id=https://api.semanticscholar.org/CorpusID:123640749#id-name=S2CID&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=752014#id-name=MR&rft_id=https://www.jstor.org/stable/3213611#id-name=JSTOR&rft_id=info:doi/10.2307/3213611&rft.au=Louchard,+G.&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Pitman, J. W. (1983). "Remarks on the convex minorant of Brownian motion". Seminar on Stochastic Processes, 1982. Progr. Probab. Statist. Vol. 5\. Birkhauser, Boston. pp. 219–227\. MR 0733673.219-227&rft.pub=Birkhauser,+Boston&rft.date=1983&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=733673#id-name=MR&rft.au=Pitman,+J.+W.&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">
* Revuz, Daniel; Yor, Marc (2004). Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften. Vol. 293\. Springer-Verlag, Berlin. doi:10.1007/978-3-662-06400-9. ISBN 978-3-642-08400-3. MR 1725357.
* Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion". Annals of Probability. 7 (1): 143–149\. doi:10.1214/aop/1176995155. JSTOR 2242845. MR 0515820.143-149&rft.date=1979&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=515820#id-name=MR&rft_id=https://www.jstor.org/stable/2242845#id-name=JSTOR&rft_id=info:doi/10.1214/aop/1176995155&rft.au=Vervaat,+W.&rft_id=https://doi.org/10.1214%2Faop%2F1176995155&rfr_id=info:sid/en.wikipedia.org:Brownian+excursion" class="Z3988">

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