Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz.\spacefactor =1000 Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright's constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.


Introduction
This survey started as an attempt to better understand two different results, viz. the asymptotic enumeration of connected graphs with a given number of independent cycles by Wright [58; 59] and the formulas for moments of the Brownian excursion area by Louchard [34], Takács [44] and others, and the surprising connection between these two seemingly unrelated results found (and explained) much later by Spencer [43], who showed that the same sequence of constants appears in both results (see (36) below). The literature may seem confusing (to me at least), however, because different authors, and sometimes the same author in different papers, have used not only different notations for the same constants but also several different related constants. I therefore set out to collect the various definitions and notations, and to list the relations between them explicitly. I was further inspired by the paper by Flajolet and Louchard [17], studying further related properties of the distribution.
After doing this, I realized that many of the results for Brownian excursion area have close parallels for the integrals of (the absolute value of) other related processes, viz. Brownian bridge, Brownian motion, Brownian meander and Brownian double meander, and also for the positive parts of Brownian motion and Brownian bridge (see . In particular, there are similar recursion formulas for moments and similar expressions for double Laplace transforms involving the Airy function, but the details differ between the seven different Brownian areas. I therefore decided to collect various results for these seven different areas of Brownian processes together so that both the similarities and the differences would become clear. One important source of results and inspiration for this is the series of papers by Takács [44;45;46;47;48;49;50]; another, treating different cases in a unified way, is Perman and Wellner [38]. In the course of doing this, I have also added some minor new results motivated by parallel results. For example, much of Section 29 and the related Appendix B, which both generalize results by Flajolet and Louchard [17], seem to be new, as do many of the explicit asymptotic results as x → 0.
To faciliate comparisons and further studies in the literature, I have often listed the notations used by other authors to provide a kind of dictionary, and I have tried to give as many explicit relations as possible, at the cost of sometimes perhaps being boring; I hope that the reader that gets bored by these details can skip them. I have found draft versions of this survey very useful myself, and I hope that others will find it useful too.
There are two main methods to prove the results on moments for the various processes given here. The first studies Brownian motion using some form of the Feynman-Kac formula, see for example Kac [29;30;31], Louchard [34], Takács [49] and Perman and Wellner [38]; this leads to various expressions involving the Airy function. See also Section 27 for a probabilistic explanation of the two slightly different forms of formulas that arise for the different cases. A related method is the path integral method used in mathematical physics and applied to Brownian areas by Majumdar and Comtet [35]. The second main method is combinatorial and studies the corresponding discrete structures, combined with singularity analysis for the resulting generating functions; see for example Takács [44;45;46;47;48;50]; and Nguyên Thê [36].
We have omitted most of the proofs, but we include or sketch some. In particular, we give for completeness in Appendix C proofs by the Feynman-Kac formula of the basic analytic identities (a double Laplace transform that is expressed using the Airy function) for all types of areas of Brownian processes studied here.
There are, of course, many related results not covered here. For example, we will not mention integrals of the supremum process or of local time [26], or processes conditioned on their local time as in [8]. See also the impressive collection of formulas for Brownian functionals by Borodin and Salminen [7].
The Brownian excursion area is introduced in Section 2 and some equivalent descriptions are given in Section 3. The results on graph enumeration are given in Section 4, the connection with the Brownian excursion area is given in Section 5, and various aspects of these results and other related results on the Brownian excursion area are discussed in Sections 6-18. Integrals of higher powers of a Brownian excursion are discussed briefly in Section 19. The other types of Brownian processes are studied in Sections 20-25; these sections are intended to be parallel to each other as much as possible. Furthermore, in Sections 24 and 25, results are given both for the positive parts of a Brownian bridge and motion, respectively, and for the joint distribution of the positive and negative parts. Sections 26 and 27 give more information on the relation between the seven different Brownian areas (the reader may prefer to read these sections before reading about the individual variables), and Section 28 compares them numerically. Finally, Section 29 discusses negative moments of the Brownian areas.
The appendices contain some general results used in the main part of the text.
The first few moments of the seven different Brownian areas studied here are given explicitly in 11 tables throughout the paper together with the first elements in various related sequences of constants. Most of these values, and some other expressions in this paper, have been calculated using Maple.
We let B(t) be a standard Brownian motion on the [0, ∞) with B(0) = 0 (usually we consider only the interval [0, 1]). We remind the reader that a Brownian bridge B br (t) can be defined as B(t) on the interval [0, 1] conditioned on B(1) = 0; a (normalized) Brownian meander B me (t) as B(t) on the interval [0, 1] conditioned on B(t) ≥ 0, t ∈ [0, 1], and a (normalized) Brownian excursion B ex (t) as B(t) on the interval [0, 1] conditioned on both B(1) = 0 and B(t) ≥ 0, t ∈ [0, 1]. These definitions have to be interpreted with some care, since we condition on events of probability 0, but it is well-known that they can be rigorously defined as suitable limits; see for example Durrett, Iglehart and Miller [14]. There are also many other constructions of and relations between these processes. For example, see Revuz and Yor [39, Chapter XII], if T > 0 is any fixed time (for example T = 1), and g T := max{t < T : B(t) = 0} and d T := min{t > T : B(t) = 0} are the zeros of B nearest before and after T , then conditioned on g T and d T , the restrictions of B to the intervals [0, g T ], [g T , T ] and [g T , d T ] are, respectively, a Brownian bridge, a Brownian meander and a Brownian excursion on these intervals; the normalized processes on [0, 1] studied in this paper can be obtained by standard Brownian rescaling as g −1/2 T B(tg T ), (T − g T ) −1/2 B(g T + t(T − g T )) and (d T − g T ) −1/2 B(g T + t(d T − g T )). (See also Section 27, where we consider Brownian motion, bridges, meanders and excursions on other intervals than [0,1], and obtain important result by this result and other rescalings.) Another well-known construction of the Brownian bridge is B br (t) = B(t) − tB(1), t ∈ [0, 1]. Further, the (normalized) Brownian excusion is a Bessel(3) bridge, i.e., the absolute value of a three-dimensional Brownian bridge, see Revuz and Yor [39].
While the Brownian motion, bridge, meander and excursion have been extensively studied, the final process considered here, viz. the Brownian double meander, has not been studied much; Majumdar and Comtet [35] is the main exception. In fact, the name has been invented for this paper since we have not seen any given to it previously. We define the Brownian double meander by B dm (t) := B(t) − min 0≤u≤1 B(u); this is a non-negative continuous stochastic process on [0, 1] that a.s. is 0 at a unique point τ ∈ [0, 1], and it can be regarded as two Brownian meanders on the intervals [0, τ ] and [τ, 1] joined back to back (with the first one reversed), see Section 23.

Brownian excursion area and its moments
Let B ex denote a (normalized) Brownian excursion and the Brownian excursion area. Two variants of this are studied by Flajolet, Poblete and Viola [18] and Flajolet and Louchard [17] (their definition is actually by the moments in (8) below), and used in [21; 23; 22]. (Louchard [34] uses ξ for our B ex ; Takács [44; 45; 46; 47; 48] uses ω + ; Perman and Wellner [38] use A excur ; Spencer [43] uses L. Flajolet and Louchard [17] use simply B for B ex , but since we in later sections will consider areas of other related processes too, we prefer to use B ex throughout the paper for consistency.) Flajolet and Louchard [17] call the distribution of A the Airy distribution, because of the relations with the Airy function given later. (But note that the other Brownian areas in this paper also have similar connections to the Airy function.) This survey centres upon formulas for the moments of B ex and various analogues of them. The first formula for the moments of B ex was given by Louchard [34] (using formulas in [33]), who showed (using β n for E B n ex ) where γ k satisfies Takács [44; 45; 46; 47; 48] gives the formula (using M k for E B k ex ) where K 0 = −1/2 and (See also Nguyên Thê [36], where E B r ex is denoted M D r and a D r equals 2 5/2−r/2 K r .) We will later, in Section 16, see that both the linear recurrence (4)-(5) and the quadratic recurrence (6)-(7) follow from the same asymptotic series (98) involving Airy functions. We will in the sequel see several variations and analogues for other Brownian areas of both these recursions.
Flajolet, Poblete and Viola [18] and Flajolet and Louchard [17] give the formula (and use it as a definition of the distribution of A; they further use µ (k) and µ k , respectively, for E A k ).
By a special case of Janson [21,Theorem 3.3], which is equivalent to (6) by (3) and (12). Further, (4) is equivalent to (8) and It is easily seen from (9) that 2 k Ω k is an integer for all k ≥ 0, and thus by (15), γ k is an integer for k ≥ 0.

Other Brownian representations
Let B br denote a Brownian bridge. By Vervaat [54], B br (·) − min [0,1] B br has the same distribution as a random translation of B ex , regarding [0, 1] as a circle.
(That is, the translation by u is defined as B ex (⌊· − u⌋).) In particular, these random functions have the same area: This equivalence was noted by Takács [44; 45], who considered the functional of a Brownian bridge in (16) in connection with a problem in railway traffic. Darling [12] considered the maximum (denoted G by him) of the process Since B br is symmetric, B br d = −B br , we obtain from (16) also Thus Darling's G := max t Y (t) [12] equals B ex (in distribution). Y (t) is clearly a continuous Gaussian process with mean 0, and a straightforward calculation It follows that if we regard [0, 1] as a circle, or if we extend Y periodically to R, Y is a stationary Gaussian process, as observed by Watson [56]. Furthermore, by construction, the integral 1 0 Y (t) dt vanishes identically.

Graph enumeration
Let C(n, q) be the number of connected graphs with n given (labelled) vertices and q edges. Recall Cayley's formula C(n, n− 1) = n n−2 for every n ≥ 1. Wright [58] proved that for any fixed k ≥ −1, we have the analoguous asymptotic formula C(n, n + k) ∼ ρ k n n+(3k−1)/2 as n → ∞, for some constants ρ k given by with other constants σ k given by σ −1 = −1/2, σ 0 = 1/4, σ 1 = 5/16, and the quadratic recursion relation Note the equivalent recursion formula Wright gives in the later paper [59] the same result in the form (although he now uses the notation f k = ρ k ; we have further corrected a typo in [59, Theorem 2]), where d 1 = 5/36 and See also Bender, Canfield and McKay [5, Corollaries 1 and 2], which gives the result using the same d k and further numbers w k defined by w 0 = π/ √ 6 and so that (Wright [59] and Bender et al. [5] further consider extensions to the case k → ∞, which does not interest us here.) In the form Next, define c k , k ≥ 1, as in Janson, Knuth, Luczak and Pittel [24, §8]; c k is the coefficient for the leading term in an expansion of the generating function for connected graphs (or multigraphs) with n vertices and n + k edges. (Note that c k = c k0 =ĉ k0 in [24, §8]. c k is denoted c k,−3k in Wright [58] and b k in Wright [59].) We have by Wright [59, §5], or by comparing (25) and (32) below, From [24, (8.12)] (which is equivalent to Wright [58, (7)]) follows the recursion where jc j is interpreted as 1/6 when j = 0. In other words, c 1 = 5/24 and By (29), (30) is equivalent to (and also for k = 0 with the interpretation 0c 0 = 1/6 again). By [24, § §3 and 8], (20) holds for k ≥ 1 with which clearly is equivalent to (21) and (24) by (33) and (30). Finally, we note that (20) can be written

Numerical values
Numerical values for small k are given in Table 1

Semi-invariants
Takács [44] considers also the semi-invariants (cumulants) Λ n of B ex , given recursively from the moments M k := E B k ex by the general formula For example, Λ 1 = E B ex = π/8, Λ 2 = Var(B ex ) = (10 − 3π)/24 and 2π/128. Since these do not have a nice form, they will not be considered further here.

The Airy function
There are many connections between the distribution of B ex and the transcendental Airy function; these connections have been noted in different contexts by several authors and, in particular, studied in detail by Flajolet and Louchard [17]. We describe many of them in the following sections.
The Airy function Ai(x) is defined by, for example, the conditionally convergent integral Ai(x) is, up to normalization, the unique solution of the differential equation Ai ′′ (x) = xAi(x) that is bounded for x ≥ 0. Ai extends to an entire function. All zeros of Ai(x) lie on the negative real axis; they will appear several times below, and we denote them by a j = −|a j |, j = 1, 2, . . . , with 0 < |a 1 | < |a 2 |, . . . . In other words, We have the asymptotic formula a j ∼ −(3π/2) 2/3 j 2/3 , which can be refined to an asymptotic expansion [1, 10.4.94], [17].  [17] and Majumdar and Comtet [35] use α j for |a j |; Darling [12] denotes our |a j | by σ j ; his α j are the zeros of J 1/3 (x) + J −1/3 (x), and are equal to 2 3 |a j | 3/2 in our notation, see (76).) We will later also need both the derivative Ai ′ and the integral of Ai. It seems that there is no standard notation for the latter, and we will use

Laplace transform
Let ψ ex (t) := E e −tBex be the Laplace transform of B ex ; thus ψ ex (−t) := E e tBex is the moment generating function of B ex . It follows from (53) that ψ ex (t) exists for all complex t, and is thus an entire function, with [34], G(2 −3/2 t) by Flajolet and Louchard [17] and Ψ e (t) by Perman and Wellner [38].) Darling [12] found (in the context of (18)) the formula, with |a j | as in (77), also found by Louchard [34] using the formula, for x ≥ 0, proved by Louchard [33]. We give a (related but somewhat different) proof of (81) in Appendix C.4. The relation (80) follows easily by Laplace inversion from (81) and the partial fraction expansion, see [17], A proof of (80) by methods from mathematical physics is given by Majumdar and Comtet [35].
Taking the derivative with respect to x in (81), we find, for x > 0, or, by the changes of variables  [17] use W and w for the distribution and density functions

Series expansions for distribution and density functions
Since B ex > 0 a.s., we consider in this section x > 0 only. Let, as above, a j denote the zeros of the Airy function. Darling [12] found by Laplace inversion from (80) (in our notation) where p is the density of the positive stable distribution with exponent 2/3, normalized to have Laplace transform exp(−t 2/3 ); in the notation of Feller [15, Section XVII.6] x k is the density of a spectrally negative stable distribution with exponent 3/2. (Takács [44] defines the function g(x) = 2 −1/3 p(2 −1/3 x; 3 2 , − 1 2 ); this is another such density function with a different normalization. Takacs [49] uses g for p.) The stable density function p can also be expressed by a Whittaker function W or a confluent hypergeometric function U [1, 13.1], [32, 9.10 and 9.13.11] (where U is denoted Ψ), Hence (85) with

Asymptotics of distribution and density functions
Louchard [34] gave the two first terms in an asymptotic expansion of f ex (x) and the first term for F ex (x) as x → 0; it was observed by Flajolet and Louchard [17] that full asymptotic expansions readily follow from (92) (for f ex ; the result for F ex follows similarly from (90), or by integration); note that only the term with j = 1 in (92) is significant as x → 0 and use the asymptotic expansion of U given in e.g. [32, (9.12.3)]. The first terms are (correcting typos in [34] and [17]), as x → 0, As in all other asymptotic expansions in this paper, we do not claim here that there is a convergent infinite series on the right hand side; the notation (using ∼ instead of =) signifies only that if we truncate the sum after an arbitrary finite number of terms, the error is smaller order than the last term. (Hence, more precisely, the error is of the order of the first omitted term.) In fact, the series in (93) and (94) diverge for every x > 0 because the asymptotic series for U does so.

Airy function expansions
Following Louchard [34], we expand the left hand side of (81) in an asymptotic series as x → ∞, using (79), and find after replacing 2 1/3 x by x the asymptotic series Note that the infinite sum in (98) diverges by (53); cf. the discussion in Section 15. For the proof of (98), we thus cannot simply substitute (79) into (81); instead we substitute a truncated version (a finite Taylor series) where N is finite but arbitrary. In similar situations for other Brownian areas in later sections, for example with (133) below for the Brownian bridge, this yields directly a sum of Gamma integrals and an asymptotic expansion of the desired type. In the present case, however, we have to work a little more to avoid divergent integrals. One possibility is to substitute (99) into the differentiated version (84), evaluate the integrals, subtract x −1/2 from both sides, and integrate with respect to x. Another possibility is to write the left hand side of (81) as and subsitute (99) into the third integral, noting that the first integral is − √ 2 x 1/2 and the second is a constant (necessarily equal to 2 1/3 Ai ′ (0)/Ai(0)).
Using (6), we can rewrite (98) as This is also, with a change of variables, given in Takács [45]. Flajolet and Louchard [17] give this expansion with the coefficients (−1) k Ω k /(2 k k!), which is equivalent by (10). They give also related expansions involving Bessel and confluent hypergeometric functions with coefficients Ω k /k!. An equivalent expansion with coeficients 3(k − 1)c k−1 , which equals Ω k /k! by (41), was given by Voblyȋ [55], see [24, (8.14) and (8.15)]. The Airy function and its derivative have, as x → ∞, the asymptotic expansions [1, 10.4.59 and 10.4.61] (we write c ′ k and d ′ k instead of c k and d k to avoid confusion with our c k and d k above) where These asymptotic expansions can also be written using e k or α k , since, by (64), (71) and (104), We thus obtain from (101)-(103), after the change of variables z = −x −3/2 ր 0, the equality for formal power series These asymptotic expansions can also be written as hypergeometric series. We have, as is easily verified, the equalities for formal power series Hence, (102) and (103) can be written, for x → ∞, and (108) can be written Flajolet and Louchard [17] give the equivalent formula, see (10), By the asymptotic expansion for Bessel functions [32, (5.11.10)] this can, as in Flajolet and Louchard [17], also be written (with arbitrary signs) The coefficients in this asymptotic series can be rewritten in various ways by the relations in Sections 2 and 6, for example by (41) as 3(k − 1)c k−1 for k ≥ 2, which yields [24, (8.15)].
If we in (108) multiply by the denominator and identify coefficients, we obtain the linear recursion (70) and the equivalent recursions (65)-(69); note that e k = 3 k c ′ k and α k = (3/2) k c ′ k by (105) and (106). On the other hand, by differentiating (101) (which is allowed, e.g. because the asymptotic expansion holds in a sector in the complex plane), This gives the equation for formal power series which is equivalent to the quadratic recursion (7).
As shown in Flajolet and Louchard [17], as a simple consequence of (82), for |z| < |a 1 |, and thus the values of Λ(s) at positive integers s = 2, 3, . . . can be computed from the Taylor series of Ai, given for example in [1, 10.4.2-5]. This and (122) gives explicit formulas for the negative moments B −s ex when s is an odd multiple of 1/3, including when s is an odd integer; see Flajolet and Louchard [17]. Alternatively, these formulas follow from (340) in Appendix B. For example, see further Section 29. We have here used the standard formula Note also that if −1 < Re s < −1/2, then by (81), Fubini's theorem and (335) in Appendix B, By (122), this equals as shown directly by Flajolet and Louchard [17].

Integrals of powers of B ex
Several related results are known for other functionals of a Brownian excursion or other variants of a Brownian motion. We describe some of them in this and the following sections, emphasizing the similarities with the results above, and in particular linear and quadratic recurrencies for moments and related formulas for generating functions.
The results above for moments of B ex = 1 0 B ex (t) dt have been generalized to joint moments of the integrals 1 0 B ex (t) ℓ dt, ℓ = 1, 2, . . . , by Nguyên Thê [37] (mainly ℓ = 1 and 2) and Richard [41] (all ℓ ≥ 1). These papers show that (6) and (7) (or equivalent formulas above) extend to these joint moments, with a quadratic recursion with multiple indices. We refer to these papers for details, and remark only that if we specialize to B ex (ℓ = 1), then A r,s in Nguyên Thê [37] specializes to similarly, in Richard [41], The recursions in both [37] and [41] specialize to (7) (or an equivalent quadratic recursion).
Janson [21] has given further similar results, including a similar quadratic recursion with double indices, for (joint) moments of B ex and another functional of a Brownian excursion; see also Chassaing and Janson [23].

Brownian bridge
Let B br := 1 0 |B br (t)| dt, the integral (or average) of the absolute value of a Brownian bridge. (Shepp [42] and Rice [40] use ξ for B br ; Johnson and Killeen [28] use L; Takács [47] uses σ. Cifarelli [9] uses C(x) for the distribution function of 2 3/2 B br .) Comtet, Desbois and Texier [10] give also the alternative representation [9] and (independently) Shepp [42] found a formula for the Laplace transform ψ br (t) := E e −tB br . (See Appendix C.2 for a proof.) An equivalent result, in physical formulation, was proved by path integral methods by Altshuler, Aronov and Khmelnitsky [3], see Comtet, Desbois and Texier [10]. Shepp's version is (with ψ br denoted φ by Shepp), cf. (81): while Cifarelli's version, which actually is stated using the Bessel function K 1/3 , cf. (74), can be written for arbitrary u > 0. Clearly, (133) is the special case u = √ 2 of (134), and the two formulas are equivalent by a simple change of variables. The moments of B br the are obtained by asymptotic expansions, cf. Section 16; Cifarelli [9] considers (134) as u → 0 and Shepp [42] considers (133) as x → ∞; these are obviously equivalent. Introduce D n defined by (Takács [47] denotes E B n br by M * n ; Perman and Wellner [38] use L n for our D n , A 0 for B br and µ n for E B n br ; Nguyên Thê [36] denotes E B n me by M P n and uses Q n for our D n and a P n for 2 −1/2−n/2 D n .) We then obtain from (133), cf. (98) and the discussion after it, and thus by (102), (103), (106), (107), cf. (108), as formal power series, Cifarelli [9] uses and writes the result of the expansion as with, as formal power series (we use a ′ k and b ′ k for a k and b k in [9]) this is by (138) equivalent to (135) and (137) together with Cifarelli [9] gives, instead of a recursion relation, the solution to (140) by the formula where |B| p is given by the general determinant formula this is a general way (when b ′ 0 = 1) to write the solution to Shepp [42] gives, omitting intermediate steps, the result of expanding (133) as (using ξ for B br ) whereē n satisfies the linear recursion Hence, (146) can, as in Perman and Wellner [38] (where further, as said above, α n is denoted by γ n ), be rewritten as which also follows from (137). Takács [47] found, by different methods, a quadratic recurrence analogous to (6) and (7), namely (135) together with D 0 = 1 and This follows also by differentiating (136), which yields, as x → ∞, and thus Note further that (101) and (136) show that and thus which can be regarded as a linear recursion for D n given K n , or conversely. Explicitly, It follows easily from the asymptotics (58) that the term with i = 0 in (154) asymptotically dominates the sum of the others and thus as given by Takács [47] (derived by him from (149) without further details). Consequently, cf. (53), by (135), or alternatively by Tolmatz [51] or Janson and Louchard [25], The entire function Ai ′ has all its zeros on the negative real axis (just as Ai, see Section 12), and we denote them by a ′ j = −|a ′ j |, j = 1, 2, . . . , with 0 < |a ′ 1 | < |a ′ 2 | < . . . ; thus, cf. (77),  [29] and Johnson and Killeen [28] use δ j for 2 −1/3 |a ′ j |.) The meromorphic function Ai(z)/Ai ′ (z) has the residue Ai(a ′ j )/Ai ′′ (a ′ j ) = 1/a ′ j at a ′ j , and there is a convergent partial fraction expansion Inversion of the Laplace transform in (133) yields, as found by Rice [40], Johnson and Killeen [28] found by a second Laplace transform inversion the distribution function F br (x) := P(B br ≤ x), here written as in Takács [47] with Numerical values are given by Johnson and Killeen [28] and Takács [47].
The density function f br = F ′ br can be obtained by termwise differentiation of (160), although no-one seems to have bothered to write it out explicitly; numerical values (obtained from (159)) are given by Rice [40].
Asymptotic expansions of f br (x) and F br (x) as x → 0 follow from (160) and (102); only the term with j = 1 in (160) is significant and the first terms of the asymptotic expansions are, as x → 0, For x → ∞ we have by Tolmatz [51] and Janson and Louchard [25] asymptotic expansions beginning with Nguyên Thê [37] extended some of these results to the joint Laplace transform and moments of B br and 1 0 |B br | 2 dt. Specializing to B br only, his moment formula is with (correcting several typos in [37]) B 0,0 = 1 and Clearly, these results are equivalent to (135) and (149), with Nguyên Thê [37] gives also the relation which by (129) and (167) is equivalent to (153). Equivalently, Some numerical values are given in Table 2; see further Cifarelli [9] (m n ), Shepp [42] (E B n br ,ē n ), Takács [47] (E B n br , D n ), Nguyên Thê [37] (E B n br , B n,m ). We define D * n := 2 3n D n andē * n := 9 −nē n = 4 n n! D n (considered by Shepp [42]); note that these are integers.
Asymptotic expansions of f bm (x) and the distribution function F bm (x) as x → 0 follow from (176) using the asymptotic expansion of U given in e.g. [32, (9.12.3)], cf. Section 15; only the term with j = 1 in (176) is significant and the first terms of the asymptotic expansions are, as x → 0, For x → ∞ we have by Tolmatz [52] and Janson and Louchard [25] asymptotic expansions beginning with Takács [49] also found recursion formulas for the moments. Define L n by (Takács [49] denotes E B n bm by µ n ; Perman and Wellner [38] use K n for our L n , A for B bm and ν n for E B n bm ; Nguyên Thê [36] uses a B n for 2 −n/2 L n .) An asymptotic expansion of the left hand side of (173) yields, arguing as for (98) and (136), Recall the asymptotic expansions (102) and (103) of Ai(x) and Ai ′ (x); there is a similar expansion of AI(x) [49], see Appendix A, where β k , k ≥ 0, are given by β 0 = 1 and the recursion relation (obtained from comparing a formal differentiation of (183) with (102)) (Takács [49] further uses h k = (2/3) k β k .) Hence, (182) yields, using (103) and (107), the equality (for formal power series) Multiplying by the denominator and identifying coefficients leads to the recursion by Takács [49], Differentiation of (182) yields, using Ai ′′ (x) = xAi(x), and thus by (136) and (182) the equality  which is equivalent to L 0 = 1 and the recursion formula Takács [49] proves also the asymptotic relations, see further Tolmatz [52] and Janson and Louchard [25], Some numerical values are given in Table 3; see further Takács [49] and Perman and Wellner [38]. We define L * n := 2 3n L n ; these are integers by (190).

Brownian meander
Let B me := 1 0 |B me (t)| dt, the integral of a Brownian meander on the unit interval, and let ψ me (t) := E e −tBme be its Laplace transform.
Takács [50] and Perman and Wellner [38] give formulas equivalent to see also Appendix C.3. Define Q n by (Takács [50] denotes E B n me by M n and ψ me by Ψ; Perman and Wellner [38] use R n for our Q n , A mean for B me and ρ n for E B n me ; Nguyên Thê [36] denotes E B n me by M F n and uses a F n for 2 1/2−n/2 Q n ; Majumdar and Comtet [35] denote E B n me by a n and Q n by R n .) An asymptotic expansion of the left hand side of (193) yields, arguing as for (98), (136) and (182), The asymptotic expansions (102) and (183) of Ai(x) and AI(x) yield, using (106), the equality (for formal power series) Differentiation of (195) yields and thus, using (101), the equality which is equivalent to the recursion formula with Q 0 = 1, proved by Takács [50] by a different method, and the differential equation also given by Takács [50] (in a slightly different form, with −z substituted for z). On the other hand, multiplying by ∞ k=0 α k z k in (196) and identifying coefficients leads to another recursion by Takács [50] and Perman and Wellner [38]: Furthermore, (108), (185) and (196) yield the equality and thus the relation By (152) we obtain from (202) also and thus the relation Takács [50] proves the asymptotic relations, see also Janson and Louchard [25], Some numerical values are given in Table 4; see further Takács [50]. We define Q * n := 2 3n Q n ; these are integers by (203) (or by (199) and (12)). The meromorphic function AI(z)/Ai(z) has residue r j := AI(a j )/Ai ′ (a j ) at a j , and the partial fraction expansion where the first sum is conditionally convergent only and the second sum is absolutely convergent, since |a j | ≍ j 2/3 and |r j | ≍ j −1/6 . (Note that r j alternates in sign.) In fact, the second version follows from an absolutely convergent partial fraction expansion of AI(z)/(zAi(z)), using AI(0) = 1/3, and the first then follows easily. A Laplace transform inversion (considering, for example, the Laplace transform of √ t ψ me ( √ 2 t 3/2 )) yields the formula by Takács [50] ψ me (t) = 2 −1/6 t 1/3 √ π ∞ j=1 r j e −2 −1/3 |aj |t 2/3 , t > 0.
A second Laplace inversion, see Takács [50], yields the distribution function F me (x) of B me as, using u j = |a j | 3 /(27x 2 ) and R j = |a j |r j = |a j |AI(a j )/Ai ′ (a j ), Numerical values are given by Takács [50]. The density function f me (x) can be obtained by termwise differentiation. A physical proof of (209) by path integral methods is given by Majumdar and Comtet [35] (denoting our r j by B(α j )), who then conversely derive (193) from (209).
Asymptotic expansions of f me (x) and F me (x) as x → 0 follow from (210) and (102); only the term with j = 1 in (210) is significant and the first terms of the asymptotic expansions are, as x → 0, For x → ∞, Janson and Louchard [25] found asymptotic expansions beginning with
Let B dm := 1 0 B dm (t) dt be the Brownian double meander area. By the definition, we have the formula note the analogy with (16) for B ex and B br . As in Section 3, there are further interesting equivalent forms of this. Since B is symmetric, where is a continuous Gaussian process with mean 0 and integral identically zero; its covariance function is, by straightforward calculation, given by Let ψ dm (t) := E e −tB dm be the Laplace transform of B dm . Majumdar and Comtet [35] give the formula (using C(α j ) for our r 2 j ), where r j = AI(a j )/Ai ′ (a j ) as in Section 22, using the partial fraction expansion, cf. (208), (There are only quadratic terms in this partial fraction expansion, because AI ′ (z) = −Ai(z) and Ai ′′ (z) = zAi(z) vanish at the zeros of Ai; hence there is no constant term in the expansion of AI(z)/Ai(z) at a pole a j .) Conversely, we will prove (221) by other methods in Section 27, and then (220) follows by (222) and a Laplace transform inversion. Majumdar and Comtet [35] found, by a Laplace transform inversion of (220), the density function f dm (x) of B dm as where U is the confluent hypergeometric function and v j = 2|a j | 3 /27x 2 , cf. (91) and (92).
Asymptotic expansions of f dm (x) and the distribution function F dm (x) as x → 0 follow as in [35] from (223), again using the asymptotic expansion of U [32, (9.12.3)]; only the term with j = 1 in (223) is significant and the first terms of the asymptotic expansions are, as x → 0, For x → ∞, Janson and Louchard [25] found asymptotic expansions beginning with The weaker statement had earlier been proved by Majumdar and Comtet [35] using moment asymptotics, see (235) below. Define W n by, in analogy with (181), An asymptotic expansion of the left hand side of (221) yields, arguing as for (98), (136), (182) and (195), By (183), (102), (106) and (196), this yields the equality (for formal power series) Consequently, or, as given by Majumdar and Comtet [35] (where E B n dm is denoted by µ n ), From (232) and (207) follows the asymptotic relation and thus by (229), see also Janson and Louchard [25], Some numerical values are given in Table 5; see also Majumdar and Comtet [35]. We define W * n := 2 3n W n ; these are integers by (232).

Positive part of a Brownian bridge
and the difference is Gaussian: see also Appendix C.2. Define D + n by (Perman and Wellner [38] use L + n for our D + n and µ + n for E(B + br ) n .) An asymptotic expansion of the left hand side of (239) yields, arguing as for e.g. (98) and (136), and thus by (102), (103), (183), (106), (107), which leads to the recursion by Perman and Wellner [38] Using (108) and (137), we obtain from (242) also and the corresponding recursions Some numerical values are given in Table 6; see further Perman and Wellner [38] (but beware of typos for k = 5 and 7). We define D + * n := 2 3n D + n ; these are integers by (245).
Tolmatz [53] gives the asymptotics, see also Janson and Louchard [25] and compare (156) and (155), or, equivalently,  Table 6 Some numerical values for the positive part of a Brownian bridge.
We remark further that if n = 2m is even, then (251) and (238) yield while the sum vanishes by symmetry if n is odd.
which can be written as a recursion relation for K 2m with even indices only.
Some numerical values are given in Table 7. In particular, as found by Perman and Wellner [38], We do not know of any formula for the density function f + br or distribution function F + br , except the Laplace inversion formulas in Tolmatz [53] and Janson and Louchard [25] (which prove that f + br exists and is continuous). For x → ∞ we have by Tolmatz [53] and Janson and Louchard [25] asymptotic expansions beginning with We do not know any asymptotic results as x → 0, but the results on existence of negative moments in Section 29 suggest that f + br (x) ≍ x −1/3 . More precisely we conjecture, based on (317), that

Positive part of a Brownian motion
Note also that the difference is Gaussian: Again, there is symmetry: B − bm d = B + bm , and we concentrate on B + bm . Let ψ + bm (t) := E e −tB + bm be its Laplace transform. Perman and Wellner [38] gave (using the no- see also Appendix C.1. Define L + n by √ π E(B + bm ) 4 = 6989 40960 Table 8 Some numerical values for the positive part of a Brownian motion.
Some numerical values are given in Table 8; see further Perman and Wellner [38]. We define L + * n := 2 3n L + n ; these are integers as a consequence of (298) below.
Define L ± k,l as the coefficient of ξ k η l in the left hand side of (273). Thus which yields the recursion formula, with L ± 0,0 = 1, where δ is Kronecker's delta.
a relation between Q j and K k with even indices only. Some numerical values are given in Table 9. In particular, as found by Perman and Wellner [38], We do not know of any formula for the density function f + bm or distribution function F + bm , except the Laplace inversion formulas in Janson and Louchard [25] (which prove that f + bm exists and is continuous). For x → ∞ we have by Janson and Louchard [25] asymptotic expansions beginning with We do not know any asymptotic results as x → 0, but the results on existence of negative moments in Section 29 suggest that f + bm (x) ≍ x −2/3 . More precisely we conjecture, based on (319), that

Convolutions and generating functions
Many of the formulas above can be written as convolutions of sequences, as is done by Perman and Wellner [38]. This gives the simple formulas below, where a letter X stands for the sequence (X n ) ∞ 0 and X * Y is the sequence defined by (X * Y ) n := n k=0 X k Y n−k . Alternatively, the formulas can be interpreted as identities for formal power series, if we instead interpret X as the generating function ∞ n=0 X n z n and * as ordinary multiplication. We let 1 denote the sequence (δ n0 ) = (1, 0, 0, . . . ) with the generating function 1.
Note that the sequences denoted by various letters below, except K, all have X 0 = 1, and thus the generating functions have constant term 1; the exception is K 0 = − 1 2 , which explains why −2K occurs frequently. (It would be more consistent to give a new name to −2K n and use it instead of K n in our formulas (mutatis mutandis), but we have kept K n for historical reasons, and because all numbers K n with n ≥ 1 are positive. See also (311), which shows that (6n−2)K n is more natural for some other purposes.) In the following list of formulas, we also give references to the corresponding equations above. Many of these formulas can also be found in Perman and Wellner [38].

Randomly stopped Brownian motion
Several of the formulas above have simple interpretations in terms of a Brownian motion stopped at a random time, which has been found and used by Perman and Wellner [38].
The reason for the introduction of Z ′ is that, as is well-known and explained in greater detail by Perman and Wellner [38], if g Z := max{t ≤ Z : B(t) = 0}, the last zero of B before Z, then conditioned on g Z , the restrictions of B to the intervals [0, g Z ] and [g Z , Z] form a Brownian bridge and a Brownian meander on these intervals, and the lengths g Z and Z − g Z of these intervals both have the distribution of Z ′ . Moreover, the restrictions to these two intervals are independent (also unconditionally, with g Z and Z − g Z independent), and thus B bm (Z) = B br (g Z ) + B me (Z − g z ), where the two variables on the right hand side are independent with the distributions of B br (Z ′ ) and B me (Z ′ ), which immediately yields (291). Considering only the positive part of B, we similarly obtain (298) [38].
For B ex , the slightly different (6) can be written Hence, the proper analogue of L n , L + n , D n , D + n and Q n is (6n − 2)K n . Note that it follows from (101) that, as x → ∞, the Laplace transform appearing in (84); further, in analogy with (308)-(310), (84) is equivalent to, for ξ > 0, For B dm we similarly let B which is equivalent to (221) by a change of variables. Furthermore, and the generating function (231) is the moment generating function of √ 2 B dm interpreted as a (divergent) formal power series.

A comparison
For easy reference, we collect in Table 10 the first two moments of the various Brownian areas treated above, together with the scale invariant ratio of the second moment and the square of the first. Note that the variables B ex , B dm , B me , B br , B bm have quite small variances; if these variables are normalized to have means 1, their variances range from 0.061 to 0.325 (in this order, for which we see no intuitive reason), which means that these variables, and in particular B ex , typically do not vary much from their means.

Negative moments
We gave in Section 18 formulas for negative moments of B ex due to Flajolet and Louchard [17]. These can be generalized using the results in Appendix B.  Table 11 (including a repetition of (124) and (125)); note that (126) enables us to rewrite the values in Table 11  For B + br we see that the right hand side of (239) is finite at x = 0, but that its derivative is ∼ cx −1/2 as x → 0 for some constant c = 0. Hence, (341) with m = 1, ν = 1/2 and Ψ(x) equal to the right hand side of (239), shows that for 0 < s < 3/2, the moment E(B + br ) −1+2s/3 is finite if and only s > 1/2. Consequently, the negative moment E(B + br ) −s is finite if and only if s < 2/3, and E(B + br ) s is an analytic function in the half-plane Re s > − 2 3 . More precisely, a differentiation of (239) shows that, for this Ψ, and thus the left hand side of (341) with m = 1 has the residue √ π Ai(0)/Ai ′ (0) 2 at s = 1/2, and consequently (341) yields (317) For B + bm we see that the right hand side of (265) is ∼ cx −1/2 as x → 0 for some constant c > 0. Hence, (338) with ν = 1 and Ψ(x) equal to the right hand side of (265) shows that for 0 < s < 1, the moment E(B + bm ) −2(1−s)/3 is finite if and only s > 1/2. Consequently, the negative moment E(B + bm ) −s is finite if and only if s < 1/3, and E(B + bm ) s is an analytic function in the half-plane Re s > − 1 3 . (This also follows from the considerations in Section 27, which imply that, for s > 0, E(B + bm (Z)) −s is finite if and only if E(B + br (Z ′ )) −s = E(B + br ) −s E(Z ′ ) −3s/2 is, and E(Z ′ ) −3s/2 < ∞ if and only if s < 1/3.) More precisely, (265) shows If these hold and further X has moments of all (positive) orders, then E X −s is an entire function of s; further, (335) holds for Re s > 0 and the right hand side defines a meromorphic extension of ∞ 0 t s−1 ψ(t) dt to the complex plane. Proof. (i) ⇐⇒ (ii) by (335).
Assume first that f is continuous and has compact support, and define Then φ ∈ C 1 (R), and differentiation yields Hence also φ ′ ∈ C 1 , and thus φ ∈ C 2 . A second differentiation yields, using (342), Moreover, since f has compact support, (346) shows that if x is large enough, then φ(x) = φ − f φ + (x), and thus φ(x) is bounded for large x; similarly φ(x) is bounded for x → −∞. Consequently, φ is bounded on R.
Let X(t) and Y (t) be the stochastic processes X(t) := λt + t 0 V B x (s) ds and A straightforward application of Itô's formula, using (348), shows that and thus Y (t) is a local martingale. Moreover, since X t ≥ λt and φ and f are bounded, Y (t) is uniformly bounded, and thus a bounded martingale. Hence, where, recalling that φ is bounded and X(t) ≥ λt, The result (343) now follows by (351), (352), and (346), under our assumption that f is continuous with compact support. By a monotone class theorem, for example [20, Theorem A.1], (343) remains true for all bounded measurable f with support in a finite interval [−A, A], for any A < ∞. Fixing an x with φ − (x) = 0 and letting f (y) := sign(φ + (y))1[x < y < A], we see by letting A → ∞ that ∞ x |φ + (y)| dy < ∞. Hence φ + is integrable over each interval (x, ∞), and similarly φ − is integrable over each interval (−∞, x). It follows by dominated convergence, considering f (x)1[|x| < n] → f (x), that (343) holds for all bounded measurable f on R.
Remark C.2. Majumdar and Comtet [35] have given "physical proofs" of (80) and (209) using path integral techniques. This method is closely related to the Feynman-Kac method as formulated in Lemma C.1, which can be seen as follows, where we argue formally and ignore giving proper technical conditions on V and f for the validity of the argument.
Let H be the differential operator Hφ := − 1 2 φ ′′ + V φ. (This is an unbounded positive self-adjoint operator in L 2 (R).) Then (348) can be written Hence, the right hand side of (343), which is 2 w φ(x) by (346) which essentially is the path integral formula used by Majumdar and Comtet [35] in their proof.

C.2. Brownian bridge
Let B (t) x,y denote the Brownian bridge on [0, t] with boundary values B (t) x,y (0) = x and B (t) x,y (t) = y; it can be defined by conditioning B x on {B x (t) = y}, interpreted in the usual way (with a distribution that is a continuous function of y). The standard Brownian bridge B br equals B x,y (s)) ds f (y)e −(y−x) 2 /(2t) dy √ 2πt dt (364) while the right hand side can be written, with x ∨ y := max(x, y) and x ∧ y := min(x, y), Hence, using also Fubini in (364), x,y (s)) ds e −(y−x) 2 /(2t) dt √ 2πt for a.e. y. However, both sides of (366) are continuous functions of y (the left hand side by dominated convergence because B (t) x,y (u) d = B (t) x,0 (u)+ yu/t in C[0, t] and thus B x,y in C[0, t] as y n → y). Hence, (366) holds for every y ∈ R.