ESAIM: PS 19 (2015) 578–589 ESAIM: and DOI: 10.1051/ps/2015009 www.esaim-ps.org

SOME EXPLICIT FORMULAS FOR THE , BROWNIAN MEANDER AND UNDER UNIFORM SAMPLING

Mathieu Rosenbaum1 and Marc Yor1

Abstract. We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge given in [M. Rosenbaum and M. Yor, S´eminaire de Probabilit´es XLVI (2014) 359–375], together with various√ relationships between the laws of these four processes. Finally, we consider the variable BUT1 / T1,whereB is a Brownian motion, T1 its first hitting time of level one and U a uniform random variable independent of B. This variable is shown to be centered in [R. Elie, M. Rosenbaum and M. Yor, Electron. J. Probab. 37 (2014) 1–23; M. Rosenbaum and M. Yor, S´eminaire de Probabilit´es XLVI (2014) 359–375]. The results obtained here enable us to revisit this intriguing property through an enlargement of filtration formula.

Mathematics Subject Classification. 60G40, 60J55, 60J65.

Received November 24, 2014.

1. Introduction

Let (Bt,t≥ 0) be a standard Brownian motion, T1 its first hitting time of level one and m ≥ 0. The goal of the paper [3] was the study of some very natural Brownian functionals, namely the normalized integrals of the Brownian motion up to its first hitting time, that is the variables   T1 T1 (m) 1 | |m ˜(m) 1 | |m A = 1+m/2 Bs sgn(Bs)ds, A = 1+m/2 Bs ds. T1 0 T1 0

This led us to investigate the properties of the random variable α defined by

BUT α = √ 1 , (1.1) T1 with U a uniform random variable on [0, 1], independent of B. Indeed, it is closely connected to A(m) and A˜(m). For example, for m an odd integer, we see that the m-th moment of α is the expectation of A(m).In[3], it is

Keywords and phrases. Brownian motion, Brownian bridge, Brownian meander, pseudo-Brownian bridge, Bessel process, uniform sampling, local times, hitting times, enlargement of filtration. 1 LPMA, University Pierre et Marie Curie - Paris 6, Paris, France. [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2015 EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING 579

first shown that the random variable α is centered. Intrigued by this property, we determined the distribution of this variable, which is expressed in [8] under the following form, where = denotes equality in law: L 1 α = ΛL1 − |B1|, (1.2) L 2 with Lt the at point 0 of B at time t and Λ a uniform random variable on [0, 1], independent of (|B1|,L1). The centering property is easily recovered from (1.2)since   1 1 E ΛL1 − |B1| = E[L1 −|B1|]=0. 2 2

In fact, in [8], a preliminary to the proof of (1.2) is to obtain the law of a triplet of random variables defined in terms of the pseudo-Brownian bridge introduced in [1], see Section 2 below. In this paper, we show that the law of this triplet enables us to derive several unexpected simple formulas for various quantities related to some very classical Brownian type processes, namely the Brownian bridge, the Brownian meander and the three dimensional Bessel process. More precisely, we focus on distributional properties of these processes when sampled with an independent uniform random variable. Thus, this work can be viewed as a modest complement to the seminal paper by Pitman, see [6], where the laws of these processes when sampled with (several) independent uniform random variables are already studied. The paper is organized as follows. In Section 2, we give some preliminary results related to the law of (|B1|,L1). Indeed, they play an important role in the proofs. Distributional properties for the Brownian bridge are established in Section 3 whereas the Brownian meander and the three dimensional Bessel process are investigated in Section 4. Finally, in Section 5, we reinterpret the fact that α is centered through the lenses of an enlargement formula for the Brownian motion with the time T1 due to Jeulin, see [5]. In particular we show 2 that this centering property can be translated in terms of the expectation of the random variable U/(RU R1), where R is a three dimensional Bessel process and U a uniform random variable on [0, 1] independent of R.

2. Some preliminary results about the law of (|B1|, L1)

Before dealing with the Brownian bridge, the Brownian meander and the three dimensional Bessel process, we give here some preliminary results related to the distribution of the couple (|B1|,L1). These results will play an important role in the proofs of our main theorems. + It is well-known that the law of (B1,L1) admits a density on R × R .Itsvalueatpoint(x, l)isgivenby   1 (l + |x|)2 √ (|x| + l)exp − · (2.1) 2π 2 For l ≥ 0, we set  +∞ 2 2 H(l)=el /2 dxe−x /2. (2.2) l The following consequences of (2.1) shall be useful in the sequel.

Proposition 2.1. Let l ≥ 0. We have the double identity

E[L1||B1| = l]=E[|B1||L1 = l]=H(l). (2.3)

Furthermore, one has   1 H(l)=lE , (2.4) N 2 + l2 where N denotes a standard Gaussian random variable. 580 M. ROSENBAUM AND M. YOR

Proof. We start with the proof of (2.3). Of course, it can be deduced from (2.1) at the cost of some integrations. We prefer the following arguments. First, the equality on the left hand side of (2.3) stems from the symmetry of the law of (|B1|,L1), which is obvious from (2.1). Thus, we now have to show the second equality in (2.3). This easily follows from the identity   L1 E[φ(L1)|B1|]=E dxφ(x) , (2.5) 0 which is valid for any bounded measurable function φ.Indeed,assuming(2.5) for a moment, using the fact that

L1 = |B1|, L we may write (2.5)as  +∞  +∞  +∞ −l2/2 −x2/2 dle φ(l)E[|B1||L1 = l]= dlφ(l) dxe . 0 0 l Hence, since this is true for every bounded measurable function φ,weget  +∞ −l2/2 −x2/2 e E[|B1||L1 = l]= dxe , l which is the desired result for (2.3). It remains to prove (2.5) for a generic bounded measurable function φ. Remark that the formula  t  t φ(Lt)|Bt| = dBsφ(Ls)sign(Bs)+ dLsφ(Ls) 0 0 is a very particular case of the balayage formula (see [7], p. 261). It now suffices to take expectation on both sides of this last equality to obtain (2.5). We now give the proof of the second part of Proposition 2.1.First,notethat    +∞ E 1 −vl2 E −vN2 2 2 = dve e . N + l 0 Using the Laplace transform of N 2,weobtain    +∞ −vl2/2 E 1 e√ · 2 2 = dv (2.6) N + l 0 2 1+v Then remark that thanks to the change of variable x2 =(1+v)l2,weget  +∞  +∞ 2 2 l 2 H(l)=el /2 dxe−x /2 = dv √ e−vl /2. l 0 2 1+v This together with (2.6) gives the second part of Proposition 2.1.  3. The Brownian bridge under uniform sampling

3.1. A key result on the pseudo-Brownian bridge Before giving our theorem on the uniformly sampled Brownian bridge, we recall a result related to the pseudo- Brownian bridge established in [8], and which is the key to most of the proofs in this paper. The pseudo-Brownian bridge was introduced in [1] and is defined by   Buτ √ 1 ,u≤ 1 , τ1 EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING 581 with (τl,l>0) the inverse local time process:

τl =inf{t, Lt >l}.

This pseudo-Brownian bridge is equal to 0 at time 0 and time 1 and has the same as the Brownian motion. Thus, it shares some similarities with the Brownian bridge, which explains its name. Let U be a uniform random variable on [0, 1] independent of B. The following theorem is proved in [8]. Theorem 3.1. There is the identity in law     BUτ1 1 1 √ , √ ,LUτ1 = B1,L1,Λ , τ1 τ1 L 2 with Λ a uniform random variable on [0, 1], independent of (B1,L1).

In other words, LUτ1 is a uniform random variable on [0, 1], independent of the pair   BUτ 1 √ 1 , √ , τ1 τ1

1 which is distributed as ( 2 B1,L1). To illustrate and partially check on Theorem 3.1, we give the following corollary.

Corollary 3.2. For any y ∈ R and λ>0, one has    2 y λ E Lτ exp − τ1 =exp(−λ(2|y| +1)). 1 2

As a consequence, we get   2 y 2(y + |y|) E L |τ1 = t =(2|y| +1)exp − · τ1 t

Note that some similar formula, where τ1 is replaced by T1,ispresentedin[3] (Prop. 3.1). A proof of y ≥ −y ≥ Corollary 3.2 may be obtained from Ray–Knight’s theorem which asserts that (Lτ1 ,y 0) and (Lτ1 ,y 0) are two independent BESQ(0) processes, starting from 1. Here we give a proof based on Theorem 3.1.

Proof. Consider the measure on R defined for f non negative measurable by    λ2 I(f)=E τ1f(BUτ )exp − τ1 , 1 2 with U a uniform random variable on [0, 1], independent of B. It is easily seen that it admits the density    2 y λ y → E Lτ exp − τ1 1 2 with respect to the Lebesgue’s measure (dy)onR. On the other hand, we may write from Theorem 3.1:      2 1 B1 λ I(f)=E 2 f exp − 2 · L1 2L1 2L1 Using the well-known identity in law:

(B1,L1)=R1 (ε(1 − U),U) , L 582 M. ROSENBAUM AND M. YOR where U is a uniform random variable on [0, 1], ε a symmetric Bernoulli variable, independent of U,andR1 the value at time 1 of a BES(3) process, independent of U and ε,weobtain      2 E 1 ε 1 − − λ · I(f)= 2 2 f ( 1) exp 2 2 R1U 2 U 2R1U

It is easily shown that for μ ≥ 0,    1 μ2 E 2 exp − 2 =exp(−μ). R1 2R1 Therefore,

 1        ∞ 1 E ε 1 − −λ − | | I(f)= du 2 f 1 exp = dyf(y)exp( λ(2 y +1)), 0 u 2 u u −∞ which ends the proof of Corollary 3.2. 

We note that, conversely, we may recover the joint law of (BUτ1 ,τ1) from Corollary 3.2. For example: • The density of BUτ1 is given by   → E 1 y 1 · y Lτ1 = 2 τ1 (2|y| +1) B • The density of √Uτ1 is given by τ1   1 2 → E √ yτ1 − 2 y Lτ1 = exp( 2y ), τ1 π

BUτ1 1 √ 1 that is τ1 is distributed as 2 B . • We have    ∞ 1 1 E √ E p φ(BUτ1 ) p = dyφ(y) p+2 [(R1) ]. ( τ1) −∞ (2|y| +1) 1 1 Uτ √ 1 Consequently, conditional on B 1 = y, τ1 is distributed as (2|y|+1) R . 3.2. Some consequences for the Brownian bridge To deduce some properties of the Brownian bridge from Theorem 3.1, the idea is to use an absolute continuity relationship between the law of the pseudo-Brownian bridge and that of the Brownian bridge shown by Biane, Le Gall and Yor in [1]. More precisely, for F a non negative measurable function on C([0, 1], R), we have      Buτ1 2 1 E F √ ,u≤ 1 = E F (b(u),u≤ 1) 0 , (3.1) τ1 π λ1   x where b(u),u≤ 1 denotes the Brownian bridge and (λu,u≤ 1,x∈ R) its family of local times. Let U be again a uniform random variable on [0, 1] independent of b. The following theorem is easily deduced from Theorem 3.1 together with equation (3.1).

Theorem 3.3. For any non negative measurable functions f and g, we have          0 0 λU π 1 E f b(U),λ1 g 0 = E f B1,L1 L1 E[g(Λ)], (3.2) λ1 2 2 with Λ a uniform random variable on [0, 1], independent of (B1,L1). EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING 583   0 0 0 Thus, λU /λ1 is a uniform random variable on [0, 1], independent of the pair b(U),λ1 which is distributed according to (3.2)withg =1. The following corollary of Theorem 3.3 provides some surprisingly simple expressions for some densities and (conditional) expectations of quantities related to the Brownian bridge. Corollary 3.4. The following properties hold: 0 + • The variable λ1 admits a density on R .Itsvalueatpointl ≥ 0 is given by l exp(−l2/2). √ 0 0 Hence, λ1 has the same law as 2E,withE an exponential random variable. Therefore, λ1 is Rayleigh distributed.

0 • The density of b(U) at point y given λ1 = l is given by   y 0 2 E[λ1|λ1 = l]=(2|y| + l)exp − (2y +2|y|l) . Consequently, there is the formula  y  λ1 2 E 0 =exp(−2y ). λ1 • The density of b(U) at point y is given by  +∞ y 2 E[λ1 ]= dz exp(−z /2). 2|y| √ Thus, we have b(U)= 2E(V/2),withE an exponential variable independent of V which is uniformly distributed L on [−1, 1]. The first part of Corollary 3.4 is obviously deduced from Theorem 3.3 and is in fact a very classical result, see [1, 2, 4, 7]. We now prove the second part. Proof. Let f be a non negative measurable function. First note that    1 0 0 y 0 E f (b(U)) |λ1 = l = E duf (b(u)) |λ1 = l = dyf(y)E λ1|λ1 = l . 0 R 0 Hence the density of b(U)atpointy given λ1 = l is equal to

y 0 E λ1 |λ1 = l .

Now, let h denote the density of the couple (B1,L1) given in equation (2.1). From Theorem 3.3, we easily get 0 that the density of b(U)atpointy given λ1 = l is equal to

π lh(2y,l) √ 2 = 2πh(2y,l)exp(l2/2). 2 l exp(−l2/2) The first statement in the second part of Corollary 3.4 readily follows from equation (2.1). For the second statement, we use the fact that      y +∞ √ +∞ λ1 1 y 0 2 E 0 = E dl E[λ1|λ1 = l] l exp(−l /2) = 2π dlh(2|y|,l). λ1 0 l 0 Using the definition of h, this last expression is equal to   exp −2y2 .  The last identity in Corollary 3.4 is easily deduced from Theorem 3.3 together with Proposition 2.1.Notethat this formula can also be found in ([9], p. 400). 584 M. ROSENBAUM AND M. YOR 4. The Brownian meander and the three dimensional Bessel process under uniform sampling

In this section, we reinterpret Theorem 3.1 in terms of the Brownian meander and the three dimensional Bessel process. 4.1. The Brownian meander   We first turn to the translation of Theorem 3.1 in terms of the Brownian meander, denoted by m(u),u≤ 1 . To do so, we use an equality in law shown by Biane and Yor in [2]. More precisely we have   0 0 ((m(u),iu),u≤ 1) = (|b(u)| + λu,λu),u≤ 1 , L where iu =infmt. u≤t≤1 Thus, we can reinterpret Theorem 3.3 as follows. Theorem 4.1. For any non negative measurable functions f and g, we have        iU π 1 E f (m(U),m(1)) g = E f |B1| + ΛL1,L1 L1g(Λ) , m(1) 2 2 with Λ a uniform random variable on [0, 1], independent of (B1,L1). y Let (λ˜1 ,y≥ 0) denotes the family of local times of m at time 1. Similarly to what we have done for the Brownian bridge, we are able to retrieve from Theorem 4.1 simple expressions for the laws of m(1) and m(U). We state these results in the following corollary. Corollary 4.2. The following properties hold: • The variable m(1) is Rayleigh distributed. • The density of m(U) at point y ≥ 0 is given by  2y y 2 E[λ˜1]=2 exp(−z /2)dz. y √ Thus, we have m(U)= 2EW ,withE an exponential variable independent of W which is uniformly distributed L on [1/2, 1]. Proof. The proof of the first part of Corollary 4.2 is obvious from Theorem 4.1. We now consider the second part. Let f be a non negative measurable function. Using Theorem 4.1 together with equation (2.1), we get that  +∞  +∞     2 x E f m(U) = dx dll(x + l)e−(x+l) /2E f + Λl · 0 0 2 Now remark that    x/2+l x 1 E f + Λl = dνf(ν). 2 l x/2 Therefore, by Fubini’s theorem, we get  +∞  +∞  x/2+l   2 E f m(U) = dx dl(x + l)e−(x+l) /2 dνf(ν) 0 0 x/2   2ν  +∞ 2 = dνf(ν) dx dl(x + l)e−(x+l) /2. 0 ν−x/2 EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING 585

Thus, the density of m(U)atpointν is given by    2ν  +∞  2ν 2 2 (x/2+ν) dx dl(x + l)e−(x+l) /2 = dx exp − 0 ν−x/2 0 2  2ν 2 =2 dz e−z /2. ν This ends the proof.  In fact, as it is the case for the Brownian bridge, we can give explicit formulas for several other quantities related to the Brownian meander, for example the law of m(U)givenm(1). However, these expressions are not so simple and therefore probably less interesting than those obtained for the Brownian bridge. 4.2. The three dimensional Bessel process

Finally, let (Rt,t≥ 0) be a three dimensional Bessel process starting from 0 and

Ju =infRt. u≤t≤1 Using Imhof’s absolute continuity relationship between the law of the meander and that of the three dimensional Bessel process, see [1, 4], we may rewrite Theorem 4.1 as follows. Theorem 4.3. For any non negative measurable functions f and g, we have        JU 1 2 E f (R(U),R(1)) g = E f |B1| + ΛL1,L1 L g(Λ) , R(1) 2 1 with Λ a uniform random variable on [0, 1], independent of (B1,L1). We finally give the following corollary. Corollary 4.4. The following properties hold: • The density of R(1) at point y ≥ 0 is given by

2 y2 exp(−y2/2). π √ • R(U)= UR(1) and its density at point y ≥ 0 is given by L  2 +∞ 2 y exp(−z2/2)dz. π y • The law of R(U) given R(1) isthesameasthelawofm(U) given m(1). The first two parts of Corollary 4.4 are in fact easily deduced from basic properties of the three dimensional Bessel process. The last part is a consequence of Imhof’s relation. 5. The centering property of α revisited through an enlargement of filtration formula

In this last section, we revisit the centering property of the variable

BUT α = √ 1 , T1 whichisprovedin[3] and leads to various developments in [8]. Our goal here is to show that this result can be recovered from simple properties of the three dimensional Bessel process sampled at uniform time, together with an enlargement of filtration formula for the Brownian motion with the time T1 due to Jeulin, see [5]. 586 M. ROSENBAUM AND M. YOR

5.1. Some preliminary remarks on the uniformly sampled Bessel’s process Let U be a uniform random variable on [0, 1], independent of the considered Bessel’s process R.Westart with the two following lemmas on the conditional expectation of the uniformly sampled Bessel’s process.

Lemma 5.1. We have    1 U E [RU |R1 = r]= r + E |R1 = r · (5.1) 2 RU Consequently,      RU 1 2 U E 2 = + E 2 · R1 2 π RU R1

Lemma 5.2. We have   U E |R1 = r = H(r), RU with H defined by (2.2). Consequently,     RU U 2 E 2 = E 2 = · R1 RU R1 π

Remark that we already proved the equality   RU 2 E 2 = R1 π in [3]. This was in fact the cornerstone of our first proof of the centering property of α. In the enlargement 2 2 of filtration approach used here, we will see that instead of RU /R1, the random variable U/(RU R1)appears naturally.

5.2. Proofs of Lemmas 5.1 and 5.2 We now give the proofs of Lemmas 5.1 and 5.2.

5.2.1. Proof of Lemma 5.1 ThefirstpartofLemma5.1 follows from the identity

   1  Rt dv E |R1 = R1 + E |R1 ,t≤ 1, (5.2) t t vRv after multiplying both sides by t and integrating in t from0to1.Toshow(5.2),weusetimeinversionwith t =1/w and  Rw = wR1/w, another three dimensional Bessel process. With this notation, using the Ito’s representation of the Bessel’s process, we get  w     dt  E[Rw − R1|R1]=E  |R1 , 1 Rt from which (5.2) is easily obtained. The second statement in Lemma 5.1 readily follows. EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING 587

5.2.2. Proof of Lemma 5.2 We start with the proof of the first part of Lemma 5.2.Let   1 v ρ = E dv |R1 = r . 0 Rv Using the same time inversion trick as for the proof of Lemma 5.1 together with the , we get  +∞    +∞   dw E 1 |  dt E 1 ρ = 2  R1 = r = 2 r  , 1 w Rw 0 (1 + t) Rt  where Pr denotes the law of a Bessel’s process R starting from r. We then use the Doob’s absolute continuity relationship, that is    Xt∧T0  Pr = Wr , Ft r Ft where Wr is the Wiener’s measure associated to a Brownian motion starting at point r, X is the canonical process and T0 is the first hitting time of 0 by X (see for example [7], Chap. XI). This together with the fact that Xt∧T0 =1T0>t Xt gives   1 1 1 Er  = Wr[T0 >t]= W0[Tr >t]. Rt r r Therefore,       Tr 1 dt 1 Tr 1 EW0 EW0 E · ρ = 2 = = r 2 2 r 0 (1 + t) r 1+Tr N + r Using equation (2.4), this is equal to H(r). This ends the proof of the first part of Lemma 5.2.Usingthe expression for the density of R1 given in Corollary 4.4, the proof of the second part readily follows remarking that    +∞  +∞  +∞ U 2 −x2/2 2 −x2/2 2 E 2 = dr dxe = dxxe = · RU R1 π 0 r π 0 π 5.3. An enlargement of filtration approach to the centering property of α

We now revisit the centering property of α through an enlargement of filtration formula. Let (Ft)denotethe  filtration of the Brownian motion (Bt)and(Ft) the filtration obtained by initially enlarging (Ft)withT1.Itis  shown in [5]that(Bt)isa(Ft) semi-martingale. More precisely,   t∧T1 t∧T1 ds 1 − Bs Bt = βt − + ds , (5.3) 0 1 − Bs 0 T1 − s  where (βt)isa(Ft) Brownian motion (in particular it is independent of T1). Taking expectation on both sides of (5.3)attimet = UT1,weget       UT1 UT1 1 ds 1 1 − Bs E[α]=−E √ + E √ ds · T1 0 1 − Bs T1 0 T1 − s

Using the change of variable s = uT1 in both integrals, we get       U  U du 1 1 − BuT E −E E √ 1 · [α]= T1 − + du − 0 1 BuT1 T1 0 1 u 588 M. ROSENBAUM AND M. YOR

Since U is independent of B and uniformly distributed on [0, 1], we get        1 (1 − u) 1 1 E[α]=−E T1 du + E √ du(1 − BuT ) . − 1 0 1 BuT1 T1 0

Thus,       1 E −E v E √1 · 2 [α]= T1 dv − + 0 1 BT1(1−v) T1

We now use Williams time reversal result:    T1, 1 − BT (1−v),v≤ 1 = (γ1, (Rvγ ,v≤ 1)) , 1 L 1 where

γ1 =sup{s>0,Rs =1}·

Hence we obtain     V 1 2E[α]=−E √ + E √ , RVγ1 / γ1 T1 with V a uniform random variable on [0, 1], independent of R. From the absolute continuity relationship between the laws of  √  Rvγ1 / γ1,v≤ 1 and (Rv,v≤ 1), see [1], we get   2 V 2E[α]= − E 2 · π RV R1 Hence E[α] = 0 if and only if   V 2 E 2 = · RV R1 π From Lemma 5.2, the last equality holds. Moreover, it has been shown without the help of our previous re- sults [3, 8]. Thus, the use of the enlargement formula of [5] provides an alternative proof of the centering property of α.

6. A few words of conclusion

Together with [3, 8], this paper is our third work where various aspects of the law of

BUT α = √ 1 T1 are investigated. For example, we have considered its centering property, the explicit form of its density, which may be directly deduced from equations (1.2)and(2.1), and its Mellin transform. In the present paper, starting from the pseudo-Brownian bridge, we obtain some results relative to the Brownian bridge, the Brownian meander and the three dimensional Bessel process. Imhof type relations between these processes allow to go from one to another. EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING 589 References

[1] P. Biane, J.-F. Le Gall and M. Yor, Un processus qui ressemble au pont brownien, S´eminaire de Probabilit´es XXI (1987) 270–275. [2] P. Biane and M. Yor, Quelques pr´ecisions sur le m´eandre brownien. Bull. Sci. Math. 112 (1988) 101–109. [3] R. Elie, M. Rosenbaum and M. Yor, On the expectation of normalized Brownian functionals up to first hitting times. Electron. J. Probab. 37 (2014) 1–23. [4] J.P. Imhof, Density factorizations for Brownian motion, meander and the three-dimensional bessel process, and applications. J. Appl. Probab. (1984) 500–510. [5] T. Jeulin, et grossissement d’une filtration. In vol. 833 of Lect. Notes Math. Springer (1980). [6] J.W. Pitman, Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4 (1999) 1–33. [7] D. Revuz and M. Yor, Continuous martingales and Brownian motion. In vol. 293. Springer (1999). [8] M. Rosenbaum and M. Yor, On the law of a triplet associated with the pseudo-brownian bridge. S´eminaire de Probabilit´es XLVI (2014) 359–375. [9] G.R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics. In vol. 59. SIAM (2009).