Brownian excursion – Wikipedia

Stochastic process

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,

${displaystyle e}$

, 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 Kiyosi Itô 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]

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

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

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

Properties[edit]

Vervaat’s representation of a Brownian excursion has several consequences for various functions of

${displaystyle e}$

. In particular:

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

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

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

The following result holds:^[5]

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

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

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

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[edit]

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

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

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

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

and

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

${displaystyle A_{+}}$

and many other area functionals. In particular,

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

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]

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.
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.
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.
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.
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.
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.
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.
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”. Journal of Applied Mathematics and Stochastic Analysis. 4: 263–292. doi:10.1155/S1048953391000011.
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.

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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.