The Airy distribution: experiment, large deviations and additional statistics

The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations – the Donsker-Varadhan formalism and the optimal fluctuation method – manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area, and measure its mean in the experiment. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the FerrariSpohn distribution, and we identify the reason for this coincidence.

Brownian motion came to prominence in physics and other sciences with the theoretical works of Einstein [1], Smoluchowski [2] and Langevin [3], and the experimental work of Perrin [4]. Today, more than a hundred years since those remarkable discoveries, Brownian motion is a central paradigm in a multitude of fields [5,6]. Here we focus on some interesting properties of conditioned Brownian motions as described by the Airy distribution and by its extensions that we will introduce.
Since its discovery nearly four decades ago [7,8], the Airy distribution (AD) keeps reappearing in seemingly unrelated problems in different fields of science. One of its first applications was to inventory problems where the AD describes, for example, the distribution of the time spent by locomotives in a railway depot [9,10]. The AD appears in the description of the computational cost of data storage algorithms [11]. In graph theory the AD is the distribution of the internal length of a rooted planar tree [10]. More recently the AD appeared in physics: as the distribution of the maximal height of fluctuating interfaces [12][13][14], the avalanche size distribution in sandpile models [15], the size fluctuations of ring polymers [16], and the position distribution of laser cooled atoms [17]. See Ref. [5] for a review of some of these examples.
In spite of its importance, the AD has not yet been measured in an experiment. Here we report such a measurement in the simplest setting where the AD was originally discovered [7,8]: the area under a Brownian excursion in one dimension. We also advance the theory of the AD by focusing on its previously unnoticed large-deviation properties. We show how two different large-deviation formalisms manifest themselves in the AD. This allows us to probe a previously inaccessible important quantity: the position distribution of a Brownian excursion conditioned on a specified area.
The AD. Consider a Brownian excursion: a Brownian motion x(t), conditioned to start and end at the origin, x(t = 0) = x(t = T ) = 0, and to stay positive, x(t) > 0 for 0 < t < T . The area under the Brownian excursion, is a random variable characterized by the probability distribution P (A, T ): the AD. The only dimensional parameters entering the problem are A, T and the particle diffusivity D 0 [18]. Dimensional analysis yields The Laplace transform of the scaling function f , found by probabilistic methods [7,8], was formally inverted to give the closed analytic form [10] f (ξ) = 2 √ 6 ξ 10/3 ∞ k=1 e −β k /ξ 2 β 2/3 k U −5/6, 4/3, β k /ξ 2 , where U (. . . ) is the confluent hypergeometric function [19], β k = 2α 3 k /27, and α k are the ordered absolute values of the zeros of the Airy function Ai(ξ) [20]. The Laplace transform of f was also obtained by using path integral techniques [13]. The function f (ξ) is shown in Fig. 1.
Experiment. Our experimental setup is made of colloidal suspensions of silica spheres in water (1.50±0.08 µm in diameter, mass density of 2.0 g/cm 3 , Polysciences Lot # A762412), which are loaded into a sample cell of dimensions 22 × 22 × 0.04 mm constructed from a microscope slide and a cover-slip. The particles are then allowed to sediment and equilibrate and diffuse close to the bottom slide for 30 minutes at room temperature before measurements start. Quasi-2D monolayers of area fraction φ = 0.069 ± 0.005 and φ = 0.062 ± 0.005 are prepared by diluting the original suspension with double distilled water (DDW, 18 MΩ). Sample walls are coated with bovine serum albumin to avoid particle attachment . Symbols: experimentally measured histogram of the area under 12,240 excursions of duration T 33.3 sec. Error bars are too small to be shown. Dashed and dotted lines: the small-and large-area asymptotics, respectively. They are described by Eqs. (4) and (5), but also include pre-exponential corrections, see Ref. [21].
to the bottom wall of the cell. Particle position and motion in the plane perpendicular to the optical axis are observed using bright field microscopy (Olympus IX71). Images are captured by a CMOS camera (Grasshopper 3, Point Grey Research) at a rate of 30 fps to allow for easy particle tracking. Conventional single particle tracking techniques were used to extract particle location with an accuracy of 6 nm [22]. The particle diffusivity was evaluated from the particle's mean square displacement D 0 = ∆x 2 (t) /(2t) averaged over all particles in the ensemble. Excursions are constructed from the trajectories using the Vervaat transform, see the Supplemental Material (SM) for details [23]. Fig. 1 shows the measured histogram of the area under excursions of the colloidal particles showing good agreement with the exact analytic expression (3). The histogram is a bit broader than the theoretical distribution in the region of the maximum. This effect is explained by the small but finite polydispersity of the particle diameters, leading to small variations of their diffusivity.
Large deviations. Now we turn to theory of the AD. The expression (3) is quite complicated. As a result, calculating the moments of the AD is already challenging [24]. Here we will focus on the AD's asymptotics [25,26]: which are depicted, with account of pre-exponential factors, calculated in Ref. [21], in Fig. 1. They correspond to very small or very large values of the dimensionless pa-rameterÃ ≡ AD −1/2 0 T −3/2 which plays a prominent role in this Letter. As we show here, these asymptotics are intimately connected with two different large-deviation formalisms of statistical mechanics. Let us define the rescaled area a = A/T , which is the time-averaged position of the Brownian excursion. At fixed a, the small-A limit (4) corresponds to very long times, T a 2 /D 0 , whereas the large-A limit (5) corresponds to very short times, T a 2 /D 0 . Let us start with the small A (or long time) limit. Similarly to many other time-averaged observables, the distribution of a obeys a large deviation principle due to Donsker and Varadhan (DV) [27][28][29][30], where the longtime probability of observing any finite a decays exponentially with time: From dimensional analysis, the rate function I(a) scales as D 0 /a 2 , already reproducing the correct scaling behavior (4) of the small-A tail. Reproducing the numerical factor 2α 3 1 /27 takes slightly more effort. It boils down to determining the ground state of a Schrödinger-type "tilted operator", obtained from the generator of the constrained Brownian excursion [27,28]. We present this calculation in detail in the SM [23]. The trajectories x (t), which mostly contribute to the small-A tail (4), stay close to the origin, without ever crossing it, for a long time. As we show below, the position distribution, characterizing these trajectories, is stationary for most of the time, which explains the exponential decay of P with time, see Eq. (6).
The large-A tail (5) is markedly different. It is dominated by a single, most probable excursion which realizes the prescribed large A by straying far away from the origin during a very short time. Other trajectories with the same A have exponentially smaller probabilities. This optimal trajectory, x * A (t), can be found by the optimal fluctuation method (OFM). For the Brownian motion, the OFM becomes geometrical optics [31][32][33][34][35][36][37]. The starting point of the OFM is the path probability measure of a Brownian trajectory x(t). It is given, up to pre-exponential factors, by the Wiener's action, The optimal trajectory can be found by minimizing the action (7) along excursions x (t) subject to the constraint (1). The latter can be accommodated via a Lagrange multiplier λ, leading to an effective Lagrangian L (x,ẋ) = x 2 /2 − λx, describing the classical motion of a particle in a constant field. The optimal trajectory is a parabola, where we have imposed x(0) = x(T ) = 0 and set λ = 12A/T 3 to obey Eq. (1). x * A (t) is also the average position over all trajectories with the large prescribed A. in the limit ofÃ 1, a good agreement is observed already forÃ 1.
Plugging Eq. (8) into the action (7), we exactly reproduce Eq. (5), see also Refs. [37,39]. The large-A tail (5) is shared by other Brownian motions (such as the Brownian bridge and its absolute value [40]) which start at x = 0 at t = 0 and return to x = 0 at t = T , but are allowed to reach or even cross x = 0 at 0 < t < T . The large-A tails coincide because the optimal trajectory (8) is unaffected by the non-crossing constraint.
Position distribution conditioned on A. What is the position distribution of the excursion, This important distribution has been previously inaccessible. Methods like Doob's h-transform [41] were found intractable due to the conditioning on both X and A [42,43]. An alternative method, proposed for a generalized Brownian bridge [44][45][46], could not be extended to excursions, as it is restricted to Gaussian processes. As we show now, the two large-deviation techniques that describe the two tails of f (ξ), allow one to evaluate p A at small and large A. From simple dimensional analysis, the distribution must have the scaling form AtÃ 1, the conditional distribution can be found with the DV formalism [47], see the SM [23]. Apart from narrow temporal boundary layers at t = 0 and t = T , this formalism predicts the stationary position distributioñ where Ai is the derivative of the Airy function with respect to its argument. The first moment of the distribution (10) is equal to unity, which gives x A (t) A/T in the dimensional variables. Figure 3a shows good agreement between the distribution of simulated excursions, conditioned by small area, and Eq. (10). See also Fig. 2a for a comparison of the mean.  (14) and (15) (b). In the latter prediction the normalization factor was accounted for.
Panel (a) also shows the histogram at t/T = 3/4 (squares), confirming the time-independence of the position distribution at small areas.
The distribution (10) is known as the Ferrari-Spohn (FS) distribution. It appeared previously in the studies of position fluctuations of a Brownian excursion, conditioned to stay away from a wall x w (t) which moves, sufficiently fast, back and forth so that x w (0) = x w (T ) = 0 and x w (0 < t < T ) > 0 [48][49][50]. For a parabolic wall function, the small-A distribution (10) and the FS distribution [50] coincide. This unexpected coincidence can be explained by an exact mapping that we found between the two systems [23]. The mapping involves a biased ensemble [51]: an ensemble of excursions, where the probability of each excursion is re-weighted by the exponential fac- is the area of the excursion. This ensemble and the ensemble of excursions conditioned on A are equivalent, and they obey the same relation as the one between the canonical and microcanonical ensembles, respectively, in equilibrium statistical mechanics [51]. If we denote by p (X, t; µ) the position distribution in the biased ensemble, and by p C (X, t) the position distribution of the particle relative to the parabolic wall (11) in the FS problem, the mapping reads The FS distribution has also appeared in other systems [36,[52][53][54][55]. This is indicative of a universality class. AtÃ 1 the conditional distributionpÃ (z, t/T ) is time-dependent, and it can be found with the OFM. The probability of an excursion to reach a specified position x (t) = X and to have a given large area A comes from the optimal trajectory which minimizes the action (7) under the two constraints. The optimal trajectory consists of two parabolic segments, joined at time t with a corner singularity there [23]. For the particular case of conditioning on the position x = X at t = T /2, the optimal trajectory is This trajectory is shown, for several values of X, in Fig. 4a. However, the solution (12) is Optimal paths conditioned on x (t = T /2) = X and the area A, in the subcritical (a) and supercritical (b) regimes, see Eqs. (12) and (13) a legitimate Brownian excursion only when X is smaller than a critical value X c1 = 3a. For X > 3a, x(t ) from Eq. (12) would cross the origin, which is forbidden. The correct solution for X > 3a is provided by the tangent construction of the calculus of one-sided variations [56], and we obtain see Fig. 4b. The conditional distribution is given by ∆s: the difference between the action (7) along the additionally constrained trajectory, Eq. (12) or (13), and the action along the optimal trajectory (8) constrained by the area alone. As a result, The large deviation function has a singularity: its third derivative is discontinuous at z = 3. This singularity can be interpreted as a dynamical phase transition of the third order. Similar singularities have been recently found in other systems which involve Brownian motions pushed into large-deviation regimes by constraints [34][35][36][37].
The sub-critical result (15) describes Gaussian fluctuations around the mean valuez = 3/2 which corresponds to an excursion conditioned on A but not on X. This result is in good agreement with simulations, see Fig. 3b. In the supercritical regime, Eq. (16), the support of the optimal trajectory is shorter than T , see Fig. 4b. As a result, the time T cannot enter the final result. This, and the scaling relation (14), explain the fact that the first term in Eq. (16) [which comes from the action along the supercritical trajectory (13)] is cubic.
If the condition x (t) = X is specified at t = T /2, the solution becomes a bit more involved [23]. Here the third-order dynamical phase transition occurs at a critical line X c1 (t). Furthermore, the optimal trajectory becomes asymmetric around T /2, and an additional third-order transition occurs at a higher critical value X c2 (t) > X c1 (t) [23]. In the subcritical regime, X ≤ X c1 (t), Eqs. (14) and (15) give way to These predictions also agree with simulations, see Fig. 2 b. As the excursions start and end at x = 0, the variance vanishes at t = 0 and t = T . More surprisingly, the variance (18) has a local minimum at t = T /2 and is maximal at t = T 3 ± √ 3 /6. The appearance of the local minimum of σ 2 (t/T ) at t = T /2 is not exclusive to the large-A limit: we observed it in simulations for all values ofÃ, but it is most prominent in the large-A tail. In the subcritical regime X ≤ X c1 (t), the optimal trajectory is unaffected by the constraint at the origin for any t [23]. This explains the coincidence of the Gaussian fluctuations in this regime, Eqs. (17) and (18), with those of a Brownian bridge conditioned on A [45,46].
The two large deviation formalisms -the DV method and the OFM -can be applied to other stochastic processes and dynamical observables. One example is the distribution P (B, T ) of the area under the square of a Brownian excursion, B = A promising direction of future work is to study the statistics of time-integrated quantities in multi-particle systems. For a large number of particles such systems can be efficiently probed with still another large-deviation technique: the rapidly developing Macroscopic Fluctuation Theory [57].
In conclusion, we measured, for the first time, the Airy distribution (AD) in a direct experiment. Theorywise, we uncovered connections of the AD with two different large-deviation formalisms and with the Ferrari-Spohn distribution, argued for a generality of this type of connections in other systems, and found two dynamical phase transitions. Experiment and numerical simulations produce free Brownian trajectories, rather than Brownian excursions. We obtained the latter from the former by applying two successive transformations, see e.g. Ref. [1]. First, we employ a well-known mapping which transforms a free Brownian motion x Bm (t) into a Brownian bridge x Br (t) of duration T [2]: x Next we employ the Vervaat's transform [3] which transforms a Brownian bridge into a Brownian excursion x Ex (t). The Vervaat's transform can be realized in three steps. First, place the origin at the absolute minimum attained by the bridge, such that it is positive at all times. Next, shift the time by τ , the time at which the minimum was attained: t → t + τ . Finally, glue the first part of the trajectory from t = 0 to t = τ with the remainder of the trajectory from t = τ to t = T : This procedure yields, in experiment and simulations, an ensemble of excursions with a prescribed duration T . In order to test our theoretical predictions for the tails of the position distribution of the Brownian excursion, conditioned on a given area, we had to generate sufficiently many trajectories. This was easier to do in simulations than in experiment. In our simulations with the relatively large area under excursion we used 5, 243 trajectories whose areas fit into the windowÃ = 1.3 ± 0.0075. These were extracted from a total of 5 × 10 7 Brownian excursions unconditioned by a specified area. Each trajectory was sampled along 1, 000 equally spaced points during its dynamics. For the small-area simulations, in the windowÃ = 0.3 ± 0.0075, we used 1, 885 trajectories. These were extracted from a total of 2 × 10 6 unconditioned Brownian excursions. Each trajectory here was sampled along 5, 000 equally spaced points. We verified the simulation method by measuring the area distribution of the excursions and comparing the results with the theoretical prediction (2) and (3). Very good agreement was observed.

B. The DV formalism and the small-A tail
The DV formalism [4] reduces the problem of finding the rate function I(a), defined in Eq. (6) of the main text, in the large T (or small A) limit, to an effective eigenvalue problem. According to Gärtner and Ellis [5], the rate function I(a) of the main text is given by a Legendre-Fenchel transform of the scaled cumulant generating function (SCGF) where ... denotes averaging over values of a with respect to the Airy distribution (2). According to the DV method, where ξ max is the maximal eigenvalue of the operatorL (k) ≡L + kx, which is a tilted version of the Fokker-Planck generatorL, defined by the Fokker-Planck equation corresponding to the Langevin equation for the stochastic process in question. For Brownian excursions the Fokker-Planck generator is justL = (D 0 /2) ∂ 2 x for x > 0, andL = −∞ for x ≤ 0. It is convenient to convert the Fokker-Planck problem into an effective quantum mechanical one by considering the minus Fokker-Planck generator as an effective Hamiltonian,Ĥ ≡ −L, and look for ground state energy E min of the latter [6]. As a result, The effective Schrödinger equation for the Brownian excursion reads with the boundary conditions ψ (k) (0) = ψ (k) (∞) = 0. The (discrete) spectrum, corresponding to Eq. (B6) with k < 0, is E (k) n = (D 0 /2) 1/3 k 2/3 α n where n = 1, 2, . . . , and α n are the absolute values of the zeros of the Airy function [7]. The corresponding eigenfunctions are Therefore, the SCGF is given byĨ Plugging this result into Eq. (B1) (and again assuming k < 0), we see that the maximum is achieved for and Eq. (B1) yields the small-A asymptotic (4) of the main text.
The solution of the tilted-generator problem also provides the conditional position distribution, associated with a prescribed rescaled area a. This distribution is given by the product of the left and right principal eigenfunctions of the tilted operatorL (k) [8]. In our case the left and right eigenfunctions coincide due to the hermiticity of the tilted operator, and are both given by ψ k 1 (x). Their properly normalized product gives the conditional position distribution that appears in Eq. (10) of the main text, after the substitution of k (a) from Eq. (B8).

C. Mapping to Ferrari-Spohn model
Ferrari and Spohn (FS) [9] studied the statistics of the position, at an intermediate time t = t , of a Brownian bridge x (t), when the process is constrained on staying away from an absorbing wall, that is x (t) > x w (t), where x w (t) is a semicircle, x w (t) = Ct 1/2 (T − t) 1/2 . They also extended their results to other concave (that is, convex upward) functions. FS proved that at T → ∞, typical fluctuations of ∆X = x (t ) − x w (t ) away from the moving wall obey a universal distribution which depends only on the second derivativeẍ w (t ). This universal distribution can be represented as where Ai (. . . ) is the Airy function, α 1 = +2.338107 . . . is the magnitude of its first zero, Ai is the derivative of the Airy function with respect to its argument and = −2ẍ w (t ) /D 2 0 1/3 . Eq. (C1) is valid in the limit CT √ D 0 T , that is when the wall "pushes" the Brownian bridge into the large-deviation regime.
Remarkably, Eq. (10) of the main text, which describes the single-point distribution of a Brownian excursion conditioned on its covering a very small area A D 1/2 0 T 3/2 , coincides with the distribution (C1) with = 2α 1 / (3a). This suggests that the model studied in the main text is related to the FS model. Indeed, we now establish a formal mapping between the two models, and explain this coincidence.
The path integral that corresponds to the FS model is constrained by x (t = 0) = x (t = T ) = 0 and x (t) > x w (t), where s [x (t)] is the Wiener's action, given by Eq. (7) of the main text. Let us define y (t) = x (t) − x w (t). Rewriting Eq. (C2) in terms of y (t) (the Jacobian of this transformation is equal to 1), we obtain where y (t) are Brownian excursions, y (t = 0) = y (t = T ) = 0 and y (t) > 0, and the action is where we used integration by parts and defined which is independent of y (t). For the particular case of a parabolic wall The distribution of ∆X is given by where the C-dependence enters throughs. The constant s 0 is of no importance because its contributions cancel out. Equation (C7) is exact. In the large-deviation limit CT √ D 0 T , P C (∆X) is given by the FS distribution (C1) with = 4C We now wish to find a connection between the FS model and the model studied in the present work. Let us begin by defining the canonical or biased ensemble [10]. This is an ensemble of excursions which is unconstrained by a specified A, but where the probability of each excursion is re-weighted by the exponential factor e −µA[x(t)] , where A [x (t)] is the area of the excursion. The distribution of X in the canonical ensemble is given by where we defined the biased action Comparing Eqs. (C6) and (C7) with (C9) and (C10) we arrive at On the other hand, the canonical ensemble and the conditioned on A (or micro-canonical) ensemble are equivalent and share the same relation as the one between canonical and micro-canonical ensembles in equilibrium statistical mechanics [10]. In order to write this relation explicitly, we first consider the joint probability density P (X, A) of X = x (t ) and A. It is related to the conditional probability via where P (A) is given by the Airy distribution, see Eq.
(3) of the main text. Next we note that, up to a normalization constant, the joint probability of X and A in the canonical ensemble is simply P µ (X, A) ∝ e −µA P (X, A). As a result, the relation between the two ensembles can be written as where is the Laplace transform of P (X, A) with respect to A, and enforces the normalization condition ∞ 0 p (X; µ) dX = 1. Note that N (µ) is the Laplace transform of P (A), and it is known exactly [11,12]. Equations (C11)-(C15) provide an exact connection between the conditional probability distribution p (X|A) and the distribution P C (∆X) in the Ferrari-Spohn model with a parabolic wall.
In the limit of T → ∞ at fixed values of A/T and X (note that this limit implies a small area A D 1/2 0 T 3/2 ) the inverse Laplace transforms, that give P (X, A) and P (A) from F (X, µ) and N (µ), respectively, can be evaluated using the saddle-point approximation. This approximation is the basis of the DV formalism which we describe in Sec. B. As a result, the conditional distribution can be written as where µ * = µ * (A) is found from the solution of the DV eigenvalue problem, see Eq. (B8): Putting together Eqs. (C11), (C13), (C16) and (C17), we obtain where Now, since To remind the reader, the coincidence occurs in the limit of T → ∞ at fixed values of A/T and X. Larger deviations in the two models (when T is fixed, and we consider the large-∆X or large-X limit) behave differently, see Eqs. (14)-(16) of the main text and Ref. [13].

D. The OFM and the large-A tail
The conditional probability distribution p X, t|A √ D 0 T 3 is given by the ratio of the probabilities of the optimal Brownian excursion x(t ) realizing the large A with and without the additional constraint x(t) = X. The latter is accommodated into the OFM minimization problem via an additional Lagrange multiplier λ 2 leading to the effective is the delta-function [13]. There are three regimes of interest: the subcritical, the intermediate and the supercritical, separated by two third-order dynamical phase transitions, as we now describe.
In the subcritical regime, 0 < X ≤ X c1 , where the optimal trajectory is composed of two parabolic segments with a discontinuous derivative at t = t. For 0 ≤ t ≤ t the trajectory is given by Here we obtain a Gaussian distribution: where σ 2 (t) and x * A (t) are given by Eqs. (18) and (8), respectively, of the main text. When X exceeds X c1 , x(t ) from Eq. (D2) would cross the origin. For t < T /2 (t > T /2) this happens first at the right (left) end of the trajectory t = T . A solution crossing the origin is not allowed, and the correct solution in this, intermediate regime is given by the tangent construction of the calculus of one-sided variations [14]. This solution vanishes identically past a point τ that we now determine. For concreteness, let us assume that 0 < t < T /2. (The case T /2 < t < T is obtained from symmetry.) In this regime, X c1 ≤ X ≤ X c2 ≡ 3A/(2t), the optimal trajectory is where τ is given by This trajectory is shown by the dashed line in Fig.5.
In the supercritical regime X ≥ X c2 , the trajectory (D4) also becomes invalid, as it crosses the origin. Now this happens immediately: at t = 0. Again, the tangent construction is needed in order to find the valid optimal trajectory. This time the correct x(t) vanishes at two points along the trajectory: This expression, which is a simple extension of Eq. (13) to t = T /2, is shown by the dotted line in Fig. 5. At each of the two critical lines X = X c1 (t) and X = X c2 (t) a third-order dynamical phase transition occurs, corresponding to a jump in the third derivative of the large-deviation function with respect to X. At X c1 ≤ X ≤ X c2 the large-deviation function is given by where τ is given by (D5), while at X ≥ X c2 the large-deviation function is given by For the particular case t = T /2 the two dynamical phase transitions merge into a single third-order transition, as described in the main text.

E. Area under the square of Brownian excursion
Here we study the probability distribution P (B, T ) of the area under the square of Brownian excursion, Dimensional analysis yields the scaling form For convenience we will set D 0 = 1 and restore the D 0 -dependence in the final results. The Laplace transform of P (B, T ),P (λ, T ) = ∞ 0 P (B, T ) e −λB dB, was derived in Ref. [15] by probabilistic methods. For completeness, we present here a simpler and more physical derivation by using path integral methods. Our main focus, however, are the small-and large-B tails of P (B, T ) and their intrinsic connections to the DV method and the OFM, respectively.
The probability distribution P (B, T ) is given by a sum over all the Brownian excursions on the interval 0 < t < T , conditioned by Eq. (E1). There is a subtlety here: a Brownian particle, starting at the origin, would cross it infinitely many times immediately afterwards and would not stay positive as required. This difficulty is circumvented by introducing a cutoff: assuming that x(0) = x(T ) = > 0 and sending to zero at the end of the calculation, see e.g. Ref. [16]. That is, we defineP where ... denotes the expectation value over realizations of Brownian motions which satisfy x(0) = x(T ) = and stay positive for 0 < t < T . The expectation value is given by where the normalization constant Z( , T ) =P (0, T ) is the probability of a Brownian excursion unconstrained by B.