The Vervaat transform of Brownian bridges and Brownian motion

For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq 1\}}-f(\tau(f))$, where $\tau(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.


Introduction
In a recent work of Assaf et al [3], a novel path transform, called the quantile transform Q has been studied both in discrete and continuous time settings. Inspired by previous work in fluctuation theory (see e.g. and Wendel [42] and Port [36]), the quantile transform for simple random walks is defined as follows. For w a simple walk of length n, with increments of ±1 the quantile transform associated to w is defined by: where φ w is the quantile permutation on [1, n] defined by lexicographic ordering on pairs (w(j − 1), j), that is w(φ w (i) − 1) < w(φ w (j) − 1) or w(φ w (i) − 1) = w(φ w (j) − 1), φ w (i) ≤ φ w (j) if and only if i ≤ j.
As shown in [3], the scaling limit of this transformation of simple random walks is the quantile transform in the continuous case of Brownian motion B := (B t ; 0 ≤ t ≤ 1): where L a 1 is the local time of B at level a up to time 1 and a(t) := inf{a; Chaumont [13] extended partly the result to stable cases, Chassaing and Jason [12] to the reflected Brownian bridges case, Miermont [31] to the spectrally positive case, Fourati [23] to the general Lévy case under some mild hypotheses, Le Gall and Weill [29] to the Brownian tree case and more recently, Lupu [30] to the diffusion case. However, as far as we are aware, there has not been previous study of the Vervaat transform of an unconditioned Brownian motion B or of the Brownian bridges B λ,br ending at λ = 0.
The contribution of the current paper is to give some path decomposition result of Vervaat transform of Brownian bridges (for simplicity, call them Vervaat bridges) with non-zero endpoints. In the case of a Vervaat bridge with negative endpoint V (B λ,br ) where λ < 0, the key idea is to decompose it into two pieces, the first piece a Brownian excursion and the second piece a first passage bridge. The main result is stated as follows: Theorem 1.3 Let λ < 0. Given Z λ the first return to 0 of V (B λ,br ), whose density is given by the path is decomposed into two (conditionally) independent pieces: • (V (B λ,br ) u ; 0 ≤ u ≤ Z λ ) is a Brownian excursion of length Z λ ; • (V (B λ,br ) u ; Z λ ≤ u ≤ 1) is a first passage bridge through level λ of length 1 − Z λ . Note that Theorem 1.1 [41] is recovered as a weak limit λ → 0 of the previous theorem. The parametric density family (f Z λ ) λ<0 appears earlier in the work of Aldous and Pitman [2], Corollary 5 when they studied the standard additive coalescent. Precisely, Z λ d = where B 1 is normal distributed with mean 0 and variance 1. We also refer readers to Pitman [33], Chapter 4 for some discussion therein.
For the Vervaat bridges V (B λ,br ) which ends up with some positive value, it is easy to see that we have the following duality relation: In other words, looking backwards, we have a first piece of excursion above level λ followed by a first passage bridge. Note in addition that a first passage bridge form λ > 0 to 0 has the same distribution as a three dimensional Bessel bridge from λ to 0 (see Biane and Yor [8]). We have the following decomposition of Vervaat bridges with negative endpoints: Corollary 1.4 Let λ > 0. Given Z λ the time of last hit of λ by V (B λ,br ) strictly before 1, whose density is given by f Z λ (t) = f Z −λ (1 − t) as in (1), the path is decomposed into two (conditionally) independent pieces: • (V (B λ,br ) u ; 0 ≤ u ≤ Z λ ) is a three dimensional Bessel bridge of length Z λ starting from 0 and ending at λ; • (V (B λ,br ) u ; Z λ ≤ u ≤ 1) is a Brownian excursion above level λ of length 1 − Z λ . The rest of the paper is organized as follows. In Section 2, we provide two different proofs of Theorem 1.3, one via weak limit approach which is based on some bijection lemma proved in Assaf et al [3] and the other with resort to excursion theory that relies on results in Pitman and Yor [35].
In Section 3, we give a thorough study of V (B λ,br ) where λ = 0 using Theorem 1.3 and Corollary 1.4. We prove that such processes are not Markov (Section 3.2). However, they are semimartingales and an explicit decomposition is given (Section 3.3). We also relate these processes to some simpler ones (Section 3.1, 3.4) and study the convex minorant of such processes (Section 3.5).
In Section 4, we focus on studying the Vervaat transform of Brownian motion. We first prove that V (B) is not Markov as well (Section 4.1). Nevertheless, we show that it is a semimartingale and the semimartingale decomposition is given (Section 4.2, 4.3). Finally, we provide explicit formulae for the first two moments of the Vervaat transform of Brownian motion (Section 4.4).

Path decomposition for Vervaat bridges
The whole section is devoted to proving Theorem 1.3. First, we use a discrete approximation argument to obtain the path decomposition of V (B λ,br ) where λ < 0. Also we obtain an analog to Theorem 1.2 as a by-product. In the second part, we recover the same result via excursion theory.

Discrete case analysis
We begin with the discrete time analysis of random walk cases which is based on combinatorial principles. For a simple random walk w of length n with increments ±1, we would like to describe the law of V (w a ) := (V (w)|w(n) = a) where a < 0 having the same parity as n.
Denote τ V (w) = min{j ∈ [0, n]; w(j) ≤ w(i), ∀i ∈ [0, n]} (the first global minimum of the path) and K(w) = n − τ V (w) (distance from the first global minimum to the end of the path). Following from Theorem 7.3 in Assaf et al [3], the mapping w → (V (w), K(w)) is a bijection between walk(n), the set of simple random walks of length n and the set where k, called a helper variable, records the splitting position in the original path.
The following result turns out to be a direct consequence of this theorem related to Vervaat bridges. Lemma 2.1 w a → (V (w a ), K(w a )) forms a bijection between {w ∈ walk(n) : w(n) = a} (simple random walk bridges which end at a < 0) and the set Observe that, to each pair (v, k) in the above set, one can associate a unique triple • Z a is the first time that the path hits level −1, • f br,1 Z a is the sample path of a first passage bridge of length Z a through level −1, • f br,2 Z a is that of a first passage bridge of length n − Z a starting at −1 through a.
Remark that to different pairs (v, k), one may have the same triple (Z a , f br,1 Z a , f br,2 Z a ). We now focus on calculating explicitly the distribution of Z a by counting paths. By Lemma 2.1, the total number of the Vervaat transform paths (counting with multiplicity) is (see Chapter III of Feller [20]). Therefore, the total number of the Vervaat transform configurations (counting with multiplicity ) is Also note that every Vervaat transform configuration is counted exactly l times (by bijection lemma 2.1). Hence, Combining the above discussions, we get the following path decomposition result for discrete Vervaat bridges with negative endpoint: Theorem 2.2 Let a < 0 and have the same parity as n. Given Z a := min{j > 0; V (w a ) j = −1} (distributed as (3)), the path is decomposed into two (conditionally) independent pieces: n] is a random walk first passage bridge starting at −1 through level a of length n − Z a . The theorem provides a path decomposition of Vervaat bridges into two pieces of first passage bridges, one through level −1 and the other from −1 to a. Note that it is also possible to decompose the path slightly differently by a first piece of excursion and the second a first passage bridge through level −a. However, the distribution of the splitting position is much less explicit and thus does not make the proof any easier when passing to the scaling limit.

Continuous case: passage to weak limit
We now turn to the continuous case by appealing to invariance principles. We derive the path decomposition result from Theorem 2.2.
For λ < 0 and 0 < t < 1, let λ n ∼ λ √ n and have the same parity as n and t n := 2[ tn 2 ] + 1 be two fixed sequences. Let S λn be simple random walks of length n with increments ±1 which end at λ n , V (S λn ) be the associated discrete Vervaat bridge and Z λn := inf{j > 0 : V (S λ n ) j = −1}. Define V (S λn )(u); 0 ≤ u ≤ n to be the linear interpolation of the discrete Vervaat bridge V (S λn ).
] to a first passage bridge through level λ of length 1 − t, (conditionally) independent of the excursion.
Proof: The assertion (a) can be viewed as a variant of the results proved in Vervaat [41]. According to Theorem 2.2, given Z λn = t n , the path of V (S λn ) is split into two (conditionally) independent pieces of discrete first passage bridges. Following Bertoin et al [6] and Iglehart [25], the scaled first passage bridge through level −1 converges weakly to a Brownian excursion and the scaled first passage bridge from −1 to λ n converges weakly to a first passage bridge through level λ. This proves (b).
To prove Theorem 1.3, we need to compute the limiting distribution of Z λn = t n as n → ∞. Precisely, Using Stirling's formula, we see: and n−tn+|λn|−1 2 .
Injecting these terms in (4), we deduce the limiting distribution as n → ∞ given by (1) . By a local limit argument (see Billingsley [10], Exercise 25.10), we conclude that Z λ has density f Z λ given in (1).
The next theorem is a direct consequence of Theorem 1.3 and should be called a corollary at best. Because of its importance, however, we give it status of a theorem. Theorem 2.4 Given Z λ the length of first excursion of (V (B λ,br ) t ; 0 ≤ t ≤ 1) where λ < 0, the split position A λ := 1 − argmin t∈[0,1] B λ,br t (distance from the minimum of the original bridge path to the end) is (conditionally) independent of V (B λ,br ) and uniformly distributed on [0, Z λ ], In particular, its density is where f Z λ is given by (1).
Proof: Note that in the discrete case, given a Vervaat bridge path, the helper variable k takes values exactly in {0, ..., Z λn } where Z λn is the first time that the path returns to 0. This implies that given Z λn , the minimum position of the original bridge is uniformly distributed on [0, Z λn ]. We then obtain the results in the theorem by passing to the scaling limit. Remark: The above corollary holds true for λ ≤ 0 and the case λ = 0, i.e. Theorem 1.2 [7] is recovered as a weak limit λ → 0:

Path decomposition via excursion theory
In the current section, we provide an alternative proof of Theorem 1.3 using excursion theory. The proof relies on the decomposition of bridges at their minimum, similar to the decomposition at the maximum that appears in Pitman and Yor [35]. We begin with some notations.
Let p t (x, y) be the heat kernel: and P T 0,λ be the law of the Brownian bridge from 0 to λ of length T and P Ty x be the law of the Brownian path starting from x until the first time it hits y for y < x.
Given a distribution Q on paths of finite length, denote Q ∧ its image by time reversal. Given Q and Q two distributions on paths of finite length, Q • Q will be the distribution obtained by concatenating two independent paths, one following the distribution Q and the other the distribution Q . According to Corollary 3 in Pitman and Yor [35]: Observe that P Ty 0 can be decomposed as: We extend the definition of Vervaat's transform to continuous path with finite but arbitrary life-time: given a continuous function f on [0, T ] and τ (f ) the first time it attains its minimum, define Apply the Vervaat's operator V T to (5), we get: Take λ = 0 in (6), we obtain: Let Q T 0,0 be the law of positive Brownian excursion of length T (three dimensional Bessel bridge from 0 to 0). According to Vervaat's result [41], V T (P T 0,0 ) = Q T 0,0 . Thus, Injecting the above identity in (6), we get: By disintegrating (7) with respect to the life-time of paths, we see that V T (P T 0,λ ) is a concatenation of an excursion and a first passage bridge.
Let g t (λ) be the density of the first hit of λ by Brownian motion starting from 0: It follows from (7) that the density of the splitting point Z λ between the excursion and the first passage bridge in V (P 1 0,λ ) is: Remark: P.Fitzsimmons points out that the decomposition result is also a consequence of a local Williams decomposition, which can be found in the section 6 of [22].

Study of Vervaat bridges
In this section, we will study thoroughly the Vervaat bridges with non-zero endpoint. First, we give an alternative construction of V (B λ,br ) using length-biased sampling techniques. Next we show that such processes are not Markov with respect to their induced filtrations. Despite lack of markovianity, they are semimartingales with respect to their own filtrations and explicit decomposition formulae are given in both positive and negative endpoint cases. Moreover, we relate Vervaat bridges to drifting excursion by additional conditioning. To close the section, we study some properties of convex minorant of V (B λ,br ) where λ < 0.

Construction of Vervaat bridges via Brownian bridges
In the current part, we try to provide an alternative construction of the Vervaat bridges with negative endpoint via standard Brownian bridges (which end at 0). It is obvious that the Vervaat bridges with positive endpoint can be treated similarly by time reversal.
Let λ < 0. As seen in the last section, conditioned on Z λ the first return to 0, the process is split into B ex,Z λ an excursion of length Z λ followed by F λ,1−Z λ a first passage bridge through λ of length 1 − Z λ , independent of each other. Formally, V (B λ,br ) looks much like a standard first passage bridge (of length 1) except that it has an excursion piece placed first. Therefore, it is interesting to ask whether this process can be derived from standard first passage bridge via some simple operations.
Recall that a standard first passage bridge can be constructed via standard Brownian bridge by conditioning on its local time. Denote (F λ t ; 0 ≤ t ≤ 1) for a standard first passage bridge through λ < 0. Following from Bertoin et al [6], where L 0 t is the local time (of a Brownian bridge) at level 0 up to time t. In light of the above construction, the following theorem tells how to construct the Vervaat bridges with negative terminal value by standard Brownian bridges.
be the signed excursion interval which contains U . LetX be the process by exchanging the position of the excursion of X straddling time U and the path along [0, G U ], namely: Then we have the following identity in law: Proof: According to Theorem 1.3, the law of V (B λ,br ) is uniquely determined by that of the triple (Z λ , B ex,Z λ , F λ,1−Z λ ). It suffices to prove that the law of the process on the left hand side of (10) is entirely characterized by the same triple. Following Theorem 3.1 in Perman et al [32] and the discussion below Lemma 4.10 of Pitman [33], are independent and ∆ corresponds to the length of first excursion of X via length-biased sampling: , which is a first passage bridge through level λ of length 1 − ∆ by construction (9).

Remark:
The process X defined in the above theorem is a Brownian bridge conditioned on its local time, see Chassaing and Janson [12] for detail discussions. In addition, the proof of Theorem 3.1 in Perman et al [32] is extensively based on the concept of Palm distribution, which can be read from Fitzsimmons et al [21].

Vervaat bridges are not Markov
It is natural to ask whether the Vervaat bridges are Markov (with respect to their induced filtrations). In the case of negative endpoints, it is equivalent to ask whether the entrance law after the excursion piece is nice enough for the first passage bridge to produce Markov property. The following result gives a negative answer.
Before proving the proposition, we introduce some notations that we use in the current section and rest of the paper. For x, y > 0, denotẽ Note thatq t (x, y)y 2 dy is the transition kernel of three dimensional Bessel process and where g t (y) is the density of the first hitting at level y for Brownian motion given in (8).
Remark: The counter-example provided in the proof of Proposition 3.2 indicates that the main reason that makes Vervaat bridges with negative endpoint non-Markov is the lack of information on Z. Indeed, for s ≤ t ≤ 1, V (B λ,br ) t depends not only on V (B λ,br ) s but also on the event {Z ≤ s}.
It is well-known that the time reversal of any Markov process is still Markov. This result leads to the following corollary saying that the Vervaat bridges with positive endpoint is not Markov as well.
Now we know that the Vervaat bridges of non-zero endpoint are not Markov. Thus it is natural to ask how bad they may behave so that Markov property cannot be produced. This leads to the question that whether they are semimartingales and what is the semimartingale decomposition. The following section provides some positive answers to this question.

Semimartingale decomposition of the Vervaat bridges
This section is devoted to the semimartingale decomposition of Vervaat's bridges with both negative and positive endpoint. However, the treatments in two cases are different. The reason is that the split position in the case of negative endpoint (i.e. first return to 0) is a stopping time while the split position of the Vervaat bridges with positive endpoint (i.e. the last hit of λ > 0 strictly before 1) is not.
is a Brownian motion.
Proof: This is an easy consequence of Theorem 1.3 and the following identity due to Biane and Yor [8] where F λ,l is the first passage bridge of length l from 0 to λ and BES(3) |λ|→0,l is a three dimensional Bessel bridge from |λ| to 0.
We now deal with the semimartingale decomposition of (V (B λ,br ) t∧Z λ ; 0 ≤ t ≤ 1). The process is a Brownian excursion of length Z λ , absorbed at 0 after Z λ with density given as (1).
is a Brownian motion with respect the filtration of V (B λ,br ), stopped at time Z λ .
Proof: Let ε ∈ (0, 1). We introduce (B λ,ε t ; t ≥ 0) a Brownian motion with the starting point B λ,ε 0 having the same distribution as V (B λ,br ) ε∧Z λ . Let µ λ ε be the density of this distribution, we have for x > 0, In fact, conditional on Z λ > t and the value of V (B λ,br ) t , the path (V (B λ,br ) s∧Z λ ; ε ≤ s ≤ t) has the same distribution as a three dimensional Bessel bridge. It is the same for (B λ,ε given T λ,ε 0 > t − ε and the value of B λ,ε t−ε . The corresponding density is: .
Apply Girsanov's theorem, we obtain that (Y t ) t≥ε is a continuous martingale relative to the filtration of ( Since this holds for all ε sufficiently small, this proves the proposition. The above results provide the semimartingale decomposition of V (B λ,br ) for λ < 0.

Semimartingale decomposition of Vervaat bridges with positive endpoints
Let λ > 0. Recall from Corollary 1.4 that the transformed bridge V (B λ,br ) can be decomposed into a three dimensional Bessel bridge from 0 to λ and a positive excursion above λ. The density of the split position Z λ is given by given as (1).
For x, y ≥ 0 let Q t x,y be the law of the bridge of three dimensional Bessel bridge from x to y of length t. Let (R t ) t≥0 be a three dimensional Bessel process starting from 0.
The key idea is to show that for any t ∈ [0, 1), the law of (V (B λ,br ) s ) 0≤s≤t is absolutely continuous with respect to the law of (R s ; 0 ≤ s ≤ t), identify the corresponding density D λ t and deduce by applying Girsanov's theorem the semi-martingale decomposition of (V (B λ,br ) t ; 0 ≤ t ≤ 1).
We begin with a lemma computing the joint distribution of (R t , θ t ) in the case where the last hit at level λ has not yet been attained.
Lemma 3.7 On the event R t > λ, the joint distribution of (R t , θ λ t ) is: Proof: Let y > λ. Conditionally on R t = y, (R t−s ; 0 ≤ s ≤ t) is a Brownian first passage bridge from y to 0 and t − θ λ t is the first time it hits λ. Thus conditionally on R t = y, t − θ λ t is distributed according to Moreover, conditionally on R t = y and on the value of θ λ t , (R t−s ; 0 ≤ s ≤ t − θ λ t ) and (R θ λ t −s ; 0 ≤ s ≤ θ λ t ) are two independent Brownian first passage bridges, from y to λ and from λ to 0.
The next proposition proves that the law of (V (B λ,br ) s ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (R s ; 0 ≤ s ≤ t). We express the density D λ t as a deterministic function of t, R t and θ λ t .
Proposition 3.8 For any t ∈ [0, 1), the law of (V (B λ,br ) s ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (R s ; 0 ≤ s ≤ t). The corresponding density is: Proof: Observe that as a stochastic process, (D λ t ; 0 ≤ t < 1) is continuous and in particular there is no discontinuity as R t crosses the level λ. Let t ∈ (0, 1). We decompose the density D λ t as sum of two parts: D λ t = D 1,λ t + D 2,λ t , D 1,λ t accounting for the situation Z λ > t and D 2,λ t for the situation Z λ < t. On the event R t < λ, we have D λ t = D 1,λ t . Conditionally on Z λ > t and on the position of V (B λ,br ) t , the paths (V (B λ,br ) s ; 0 ≤ s ≤ t) is a three dimensional Bessel bridge from 0 to V (B λ,br ) t , i.e. these are the same conditional laws as the laws of (R s ; 0 ≤ s ≤ t) conditioned on the value of R t . Conditionally on Z λ > t and on the value of Z λ , the distribution of V (B λ,br ) t is: Therefore, Next we consider the case Z λ < t. Conditionally on Z λ < t and the position of Z λ and V (B λ,br ) t , the paths (V (B λ,br ) s ; 0 ≤ s ≤ Z λ ) and (V (B λ,br ) Z λ +s − λ; 0 ≤ s ≤ t − Z λ ) are independent and follow the law Q Z λ 0,λ respectively Q t− Z λ 0,V (B λ,br )t−λ . These are the same conditional laws as in Lemma 3.7. On the event Z λ < t, the joint distribution of (V (B λ,br ) t , Z λ ) is: We have then, Lemma 3.9 For any t ∈ (0, 1) and a ≥ 0: Proof: By change of variables z := 1 − s 1 − t , we get: Note that Differentiating with respect to x, we obtain: Moreover, ϕ satisfies the boundary condition ϕ(+∞) = 0. Thus, According to Lemma 3.9: Observe that Φ 2,λ is C 1 . Φ 1,λ and the partial derivative ∂ 1 Φ 1,λ are continuous as functions in (t, y). However, ∂ 2 Φ 1,λ (t, y) is not defined at y = λ: For t > 0, let where (W t ; t ≥ 0) is a standard Brownian motion starting from 0, predictable with respect the filtration of (R t ; t ≥ 0).

Lemma 3.10
For all t ∈ [0, 1) and λ > 0, Proof: Remark that we cannot just apply directly Itô's formula to Φ λ (t, R t , θ λ t ) since Φ λ is not regular enough. It is easy to check that Φ 2,λ and Φ 1,λ outside {y = λ} satisfy the PDE: Let (L λ t (R); t ≥ 0) be the local time at level λ of (R t ; t ≥ 0). Apply Itô-Tanaka's formula, and take into account the discontinuity of partial derivatives ∂ 2 at level y = λ, we get: (1−θ λ t ) is constant on the intervals of time where 0∨Φ 2,λ (t, R t ) is positive. From Theorem 4.2, Chapter VI of Revuz and Yor [38], follows that: on the support of dL λ s (R), (1 − θ λ s ) being equal to 1 − s. Finally which finishes the proof.
The next theorem gives the semimartingale decomposition of V (B λ,br ) where λ > 0. Let V λ := V (B λ,br ), then is a standard Brownian motion.
Proof: For t ∈ [0, 1), let The law of (X s ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (W s ; 0 ≤ s ≤ t), with density D λ t . From Lemma 3.10 follows that From Girsanov's theorem follows that the process: is a Brownian motion.
Notice that V (B λ,br ) (with λ < 0) also looks similar to this process except that the former always stays above the line t → λt while the latter doesn't share this property. A natural way to relate these two processes is to see whether conditioned on staying above the dragging line, the Vervaat bridge is absolutely continuous with respect to drifting excursion. First we need to justify that the conditioning event has positive probability. The next proposition provides a positive answer with an explicit formula. Proposition 3.12 ∀λ < 0, Proof: Following from Proposition 15 of Schweinsberg [40], fix x ∈ [λ, 0] we know the probability of a first passage bridge through level λ to stay above the dragging line tying x to λ: Therefore, where the first equality follows from the fact that the excursion piece is always above the dragging line and the second equality is a direct consequence of (14). Following the notations of discussion below Lemma 4.10 in Pitman [33], where h −2 is the Hermite function of index −2.
Now we know that the Vervaat bridge (with negative endpoint) conditioned to stay above the dragging line is well-defined. In addition, the law of its first return to 0 is given by: The next theorem provides a path decomposition result of Vervaat's bridge conditioned to stay above the dragging line and establishes connection to drifting excursion. Theorem 3.13 Let λ < 0. Given Z λ the length of first excursion of (V (B λ,br ) t ; 0 ≤ t ≤ 1|∀t ∈ (0, 1), V (B λ,br ) t > λt) (whose distribution density is given by (15)), the path is decomposed into two (conditionally) independent pieces: • V (B λ,br ) u ; 0 ≤ u ≤ Z λ |∀t ∈ (0, 1), V (B λ,br ) t > λt is an excursion of length Z λ ; In addition, (V (B λ,br ) t ; 0 ≤ t ≤ 1|∀t ∈ (0, 1), V (B λ,br ) t > λt) is absolutely continuous with respect to (B ex,λ↓ t ; 0 ≤ t ≤ 1). The corresponding density is: Proof: According to Proposition 11 of Bertoin [5], H is distributed as (1). Following Theorem 2.6 of Chassaing and Jason [12], conditioned on H, (B ex,λ↓ t ; 0 ≤ t ≤ H) is a Brownian excursion of length H. In addition, Proposition 4 of Schweinsberg [40] states that given H, (B ex,λ↓ t ; H ≤ t ≤ 1) is a first passage bridge of length 1 − H conditioned to stay above the line t → λ(t + H) for t ∈ (0, 1 − H), (conditionally) independent of the excursion piece. By change of measures, we obtain the same triple characterization in law.

Convex minorant of Vervaat bridges
In this part, we will study some properties of convex minorant of Vervaat bridge V (B λ,br ) where λ < 0. The convex minorant of a real-valued function (X t ; t ∈ [0, 1]) is the maximal convex function (C t ; t ∈ [0, 1]) such that ∀t ∈ [0, 1], C t ≤ X t . We refer to the points where the convex minorant equals the process as vertices. Note that these points are also the endpoints of the linear segments. See Pitman and Ross [34] and Abramson et al [1] for general background.
Similar to the computation in Proposition 3.12 , we have the explicit formula for the distribution of the last segment's slopes.
Corollary 3.14 Denote s l the slope of the last segment of the convex minorant for As discussed in Pitman and Ross [34], a standard first passage bridge can only have accumulations of linear segments at its start point (while Brownian motion has accumulations at two endpoints). However, seen in the beginning of the section, the greatest difference between the Vervaat bridges and the standard first passage bridges is the first excursion piece for the former. Then we can expect that the Vervaat bridges have almost surely a finite number of segments. Proof: We adopt a sample paths argument. Consider a sample path of Brownian bridge B λ,br where λ < 0and 1−A λ := argmin B λ,br (which is a.s. unique). Note that V (B λ,br ) t > 0 for t ∈ (0, A]. Consequently, the first vertex of the Vervaat bridge α 1 > A a.s. According to Pitman and Ross [34], there can be only a finite number of segments on [α 1 , 1] since accumulations can only happen at 0 on the restricted path B λ,br | [0,1−A] . Thus, the number of segments of the Vervaat bridges is a.s. finite.
However, we expect a stronger result regarding the number of segments:

The Vervaat transform of Brownian motion
In this section, we are devoted to studying the Vervaat transform of Brownian motion. We first prove that the process is not Markov with respect to its induced filtration. Next, (V (B) t ; 0 ≤ t ≤ 1) is shown to be a semimartingale with explicit formula. The computation is essentially based on the results in Section 3.3. Finally, we provide the mean and the variance of this process.  Proof: According to the above discussion,

V (B) is not Markov
since once it hits 0 on its path, V (B) has to end negatively. On the other hand, By comparing (16) and (17), we see that these two conditional probabilities fail to be equal, which implies that (V (B) t ; 0 ≤ t ≤ 1) is not Markov. Formally this means that we obtain the information at time 1 from some prior time, which violates the Markov property.

V (B) is a semimartingale -a conceptual approach
In general, when a process is Markov (with state space in R d ), we know sufficient and necessary conditions for it to be a semimartingale, see Cinlar et al [15]. However, we have seen in the preceding subsection that V (B) is not Markov. Therefore, whether V (B) is a semimartingale or not cannot be judged by classical Markov-semimartingale procedures.
In this section, we provide a soft argument to prove that V (B) is indeed a semimartingale with respect to its induced filtration using Denisov's decomposition for Brownian motion as well as Bichteler-Dellacherie's characterization for semimartingales.
We first recall some paths decomposition result for standard Brownian motion, which permits a characterization for the Vervaat transform. Following the notations in the introduction, A is the a.s. arcsine split (1−A := argmin t∈[0,1] B t ) for a standard Brownian motion. The following theorem is due to Denisov [18]: a) ), the path is decomposed into two independent pieces: Remark: The theorem simply says that given its a.s. minimum A, a standard Brownian motion is split into two conditional independent meanders of length A and 1 − A joint back to back. Therefore, (V (B) t ; 0 ≤ t ≤ 1) can be viewed as mixing of two independent joint back-to-back Brownian meanders with respect to arcsine distribution. Now we turn to some results in the classical semimartingale theory. Given a filtration (F t ) t , a process H is said to be simple predictable if H has a representation Denote S the collection of simple predictable processes and B = {H ∈ S : |H| ≤ 1} (the unit ball in S). For a given process, we define a linear mapping I X : S → L 0 by for H ∈ S. In fact, I X is defined as stochastic integral with respect to X for simple predictable processes.
The following theorem, proved independently by Bichteler [9] and Dellacherie [17] provides a useful characterization for semimartingales. We refer the readers to Jacod [27], Protter [37] and Rogers and Williams [39] for more details. Remark: Fundamentally, this theorem tells that the notion of semimartingale is equivalent to the notion of "good stochastic integrator" and it depends only on the law of the processes.
We now state the main theorem of the section: Theorem 4.4 (V (B) t ; 0 ≤ t ≤ 1) is semimartingale with respect to its induced filtration.
Proof: Fix H ∈ B and η > 0, Note that (V (B)|A = 1) is a standard Brownian meander and B me d = R 0→ρ where R x→y is a three dimensional Bessel bridge from x to y and ρ is Rayleigh distributed Girsanov's change of measure theorem guarantees that (V (B)|A = 1) is a semimartingale and so is (V (B)|A = 0) (see e.g. Imhof [26] and Azéma-Yor [4]). Thus, by Theorem 4.3,  Proof: Observe that I V (B)|A=a (H) = I 1 + I 2 , where We have then, .
a and note that where (Ṽ (B) t ; 0 ≤ t ≤ 1) is a standard Brownian meander and H i is FṼ τ i ∧a a -adapted ∀i. We have then Similarly, by independence of two decomposed meanders, whereĨ 2 is the stochastic integral associated to reversed Brownian meander. The following corollary states that the Vervaat bridges are also semimartingales, which provides an alternative proof of the semimartingale property for Vervaat bridges obtained in Section 3.3.
However, one can hardly derive an explicit decomposition formula using Bichteler-Dellacherie's approach. Let's explain why: a generic approach for the proof of Bichteler-Dellacherie's theorem is to find Q equivalent to P such that X is Q−quasimartingale (see e.g. Protter [37] for definition). By Rao's theorem, X is Q−semimartingale, which is also P−semimartingale by Girsanov's theorem. Note that Rao's theorem is based on Doob-Meyer's decomposition theorem, which in general does not give an explicit expression for two decomposed terms (in fact they are defined as some limiting processes).
Nevertheless, in the next subsection, we do provide an explicit semimartingale decomposition of V (B). The method is similar to that in Section 3.3.

Semimartingale decomposition of the Vervaat transform of Brownian motion
In this part, we will use extensively the notations defined in the section 3.3. We consider V (B) the Vervaat's transform of a Brownian motion on [0, 1]. By definition, V (B) 1 = B 1 a.s., there is ε > 0 such that for all t ∈ (0, ε), V (B) t > 0. Let Then P( T 0 ≤ 1) = 1 2 and more precisely T 0 follows the arcsine law 1 0<t<1 dt π t(1 − t) . Conditionally on T 0 ≤ 1 and on the value of T 0 , (V (B) t ; 0 ≤ t ≤ T 0 ) has the law Q T 0 0,0 and is independent from (V (B) t ; T 0 ≤ t ≤ 1). The joint law of (V (B) 1 , T 0 ) on the event T 0 ≤ 1 is: Thus the law of V (B) 1 conditionally on T 0 is: For the semi-martingale decomposition of (V (B), we will split the task in two: the decomposition of (V (B) t ; 0 ≤ t ≤ T 0 ) and the decomposition of (V (B) t ; T 0 ≤ t ≤ 1). We will start with the latter. Let ( M t ; t ≥ 0) be the process: is a Brownian motion Proof: The value of T 0 is considered as fixed. Let (B t ) t≥0 be a Brownian motion starting from 0 and M t := min For any t ∈ [ T 0 , 1), the law of (V (B) s ; T 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (B s ; 0 ≤ s ≤ t − T 0 ). The corresponding density is: In the above expression, we integrate with respect to the density (21) the function: which is the density corresponding to a Brownian first passage bridge from 0 to λ of length 1 − T 0 . (22) rewrites as: Apply Girsanov's theorem, we get the result of the lemma.
Next we deal with the semi-martingale decomposition of (V (B) t∧ T 0 ; 0 ≤ t ≤ 1). As an auxiliary problem we will study first the semi-martingale decomposition of a process (ξ t ; t ≥ 0) defined as follows: with probability 1 2 , ξ is a Bessel 3 process starting from 0. For t ∈ (0, 1), with infinitesimal probability , ξ is a positive excursion of length t, absorbed at 0 after time t. For any t ∈ (0, 1), the law of (V (B) s∧ T 0 ; 0 ≤ s ≤ t) is absolutely continuous with respect the law of (ξ s ; 0 ≤ s ≤ t). The following lemma is a variant of Proposition 3.5.

Proposition 4.8 Let
The process is a Brownian motion with respect the filtration of ξ, stopped at time T ξ 0 .
Let T ε 0 be the first time B ε hits 0. For any t and ε, the law of (ξ s ; ε ≤ s ≤ t) is absolutely continuous with respect the law (B ε The corresponding density is: (D ε t ; t ≥ 0) seen as a time-dependent process is continuous. In particular, there is no discontinuity at T ε 0 + ε. This follows from the fact that as y tends to 0, the convolution kernel y 2q u (0, y) 1 u>0 du is an approximation to the delta function. Since for t ∈ (0, 1), Apply Girsanov's theorem we get that (Y t ; t ≥ ε) is a continuous martingale relative to the filtration of (ξ t ; t ≥ ε) with quadratic variation (t − ε) ∧ (T ξ 0 − ε) + . Since this holds for all ε sufficiently small, this implies the lemma.
We introduce the functionals Φ(t, γ) andΦ(t, γ) where t is a time and γ a continuous path: where Φ λ is defined as (13).
For any t ∈ (0, 1), the law of (V (B) s∧ T 0 ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (ξ s ; 0 ≤ s ≤ t) with density Lemma 4.9 There are positive functions c 1 (t) and c 2 (t) bounded on intervals of form [0, 1 − ε], such that for all λ > 0, y > 0, θ ≤ t ∈ [0, 1): and In addition, for y > λ we obtain, and which permits to have the desired estimation. Proof: It is clear that the quadratic variation [Φ(·, ξ), ξ] t does not increase for t ≥ T ξ 0 . We need only to show that for a Bessel 3 process (R t ; t ≥ 0) Indeed, given any T ∈ (0, 1) and t ∈ [0, T ), the law of (ξ s ; 0 ≤ s ≤ t) on the event T ξ 0 > T is absolutely continuous with respect the law of (R s ; 0 ≤ s ≤ t). For any λ > 0 (Φ λ (t, R t , θ λ t ); 0 ≤ t < 1) is a positive martingale with mean 1. Apply Fubini's theorem, we obtain that (Φ(t, R); 0 ≤ t < 1) is a positive martingale with mean 1. Let (W t ; t ≥ 0) be the Brownian motion martingale part of (R t ; t ≥ 0). To prove (24) we need only to show that the process is a (true) martingale. Lemma 3.10 ensures that for any λ > 0 the process is a local martingale. Next we show that (26) is a (true) martingale. It suffices to bound the expectation of its supreme and dominated convergence theorem permits to conclude. According to Burkholder-Davis-Gundy inequality, ∃C > 0 such that From Lemma 3.10 and the bound of Lemma 4.9 follow that (Φ λ (t, R t , θ λ t ); 0 ≤ t < 1) is a square integrable martingale and which is integrable on (0, t) for any 0 ≤ t < 1. Moreover, by Cauchy-Schwarz's inequality, The problem of integrability may only occur at 0. However, by the bound of ∂ 2 Φ λ in Lemma 4.9, we know that E[∂ 2 Φ λ (s, R s , θ λ s ) 4 ] = O( 1 s 2 ) as s → 0. Thus the above term is also integrable on (0, t) for 0 ≤ t ≤ 1. We have proved that (26) is a (true) martingale for any λ > 0. Again by Cauchy-Schwarz's inequality, It follows that the expectation of the absolute value of the martingale (26)   Then is a Brownian motion.
Proof: The density process (D t ) 0≤t≤1 given by (23) is time-continuous. In particular it follows from lemma 4.9 that on the event T ξ 0 < 1, as t converges to T ξ 0 from below and ξ t converges to 0, Φ(t, ξ) remains bounded. Besides J t (ξ t ) tends to +∞ at T ξ 0 . Hence and D t is continuous as T ξ 0 . Using the semimartingale decomposition of (ξ t ; t ≥ 0) given by Lemma 4.8 and applying Girsanov's theorem together with Lemma 4.10 we get that the process

Expectation and variance for V (B)
In the current subsection, we provide the formulae for the first two moments of the Vervaat transform of Brownian motion.
Proposition 4.12 ∀t ∈ [0, 1], we have: The computation is based on Theorem 4.2, which is stated in the Section 4.2 as well as the following identities for standard Brownian meander, whose proof will be reported to the Appendix: Proposition 4.13 Let (B me t , t ∈ [0, 1]) be standard Brownian meander. We have: EB me t B me = 2 √ t.
Remark: One can also think of computing the expectation and the variance of the Vervaat bridges. However, we are not able to derive some explicit formulae for them except in the case of zero endpoint (correspond to Brownian excursion). Also note that the expectation as well as the variance of the Vervaat transform of Brownian motion can be obtained by discrete approximation.
Proof of Proposition 4.13: (a). We compute the expectation of standard Brownian meander along the path, which relies on the following identity found in Gradshteyn and Ryzhik [24]: .
By change of variables, we obtain: (b). We next calculate meander's second moment along the paths with the following identity also found in Gradshteyn and Ryzhik [24]: ∀a > 0, By change of variables, we get: (c). Finally we will compute EB me t B me 1 for 0 ≤ t ≤ 1.