R Bosquet
email: romain.bosquet@gmail.com

A Jullien
email: agnes.jullien@ifsttar.fr

P-O Vandanjon

M Dauvergne

F Sanchez

Eco-design model of a railway: A method for comparing the energy consumption of two project variants

Keywords: Energy, High-speed railway line, Operation phase, Construction phase, Montpellier-Perpignan, Variants

In accordance with the environmental concerns that national policies still address throughout the world, railways have been extensively studied to provide quantified indicators for assessing construction/operations practices.

It is essential to take energy consumption, into account since energy can be measured worldwide, in addition to constituting a global environmental load that is time-limited as regards resource availability and known as a discriminating criterion in comparing transport infrastructure. This article introduces an innovative, generic and systemic method dedicated to determining the energy consumption of a railway line during the pre-project phase by taking into account the complete life cycle of the rail infrastructure, including construction, maintenance and operations. The method developed (called « PEAM ») focuses on assessing project variants during the design stage and therefore integrates both the geometric longitudinal constraints of the line and the thicknesses/volumes over the entire itinerary as design parameters for input into the various construction scenarios. PEAM combines methodologies stemming from Life Cycle Assessment with a consumption model derived from physical modeling. The models associated with this method are then applied to study the energy consumption of a new high-speed line located in France that also has major implications for the European connections currently under investigation as well. Two project variants are compared in terms of total energy for a 50-year service life and a given characteristic rail traffic, including passenger and freight flows. Results obtained reveal a 30 % difference between the two variants, which prior to applying PEAM were considered to be relatively similar.

Introduction

At the global scale, approximately 30 % of end-user energy and 62 % of end-user oil is consumed by the transport sector, according to International Energy Agency (2013). Reducing the world's fuel consumption is one of the highest priorities for all countries due to energy security, greenhouse gas and economic concerns. In this context, high-speed trains (HST) offer many advantages, with an energy consumption significantly less than either road or air transport. According to [START_REF] Akerman | The role of high-speed rail in mitigating climate change -the swedish case europabanan from a life cycle perspective[END_REF], high-speed trains consume roughly 4 times less energy than road transport and 9 times less than the air sector (expressed in kilowatt-hours per passenger-kilometer -kWh/pkm). This, among others reasons, explains why the network of high-speed lines (HSL) is still growing at a very fast pace. According to [START_REF] Uic | High speed lines in the World[END_REF], some 21,365 km of high-speed lines are in operation across the world, 13,964 km are under construction and another 16,347 km are planned. The HSL network is expected to cover 51,677 km by 2025.

Once it has been decided to link two cities via an HSL, the facility owner conducts an assessment of the project variants. Choosing among the various possible solutions entails a complex process that includes a range of actors (government, local authorities, local residents groups, ecological associations, etc.) and criteria (economic, environmental, technical). A clearer understanding of the tradeoffs requires suitable tools to compare the different assumptions. Since the 2000's, the environmental impacts of railways have been investigated using life cycle assessments (LCA), whereas the energy and greenhouse gas criteria are currently being evaluated.

Valuable research has previously been conducted to compare various modes of transportation from an environmental perspective (e.g. [START_REF] Chester | Environmental assessment of passenger transportation should include infrastructure and supply chains[END_REF][START_REF] Hoffrichter | Rail freight in 2035 -traction energy analysis for high-performance freight trains[END_REF][START_REF] Raghunathan | Aerodynamics of high-speed railway train[END_REF][START_REF] Janic | High-speed rail and air passenger transport: a comparison of the operational environmental performance[END_REF]. These works unfortunately did not take into account the impacts of infrastructure (e.g. see [START_REF] Janic | High-speed rail and air passenger transport: a comparison of the operational environmental performance[END_REF].

Works on HST are uncommon since conventional trains are typically studied (e.g. [START_REF] Chester | Environmental assessment of passenger transportation should include infrastructure and supply chains[END_REF]. Recent findings by [START_REF] Chester | Environmental assessment of passenger transportation should include infrastructure and supply chains[END_REF] have provided specific values for HSL and other transportation modes. These results however cannot be applied to distinguish variants since only the project length is being considered. Other works compare different train types (urban transit, conventional, high-speed), such as [START_REF] Garcia | High speed, energy consumption and emissions[END_REF] and [START_REF] Lee | A study of the train performance simulation for korean next generation high-speed train[END_REF], although it is impossible to compare a given alternative using the same train technology. [START_REF] Van Wee | Environmental impacts of high-speed rail links in cost-benefit analyses: a case study of the dutch zuider zee line[END_REF] performed breakthrough research on evaluating alternatives in the Netherlands. Different rail links with dissimilar trains have been studied. Regrettably, the method adopted to estimate the indirect energy (resulting from the infrastructure impact) is only briefly described and limited to freight infrastructure. Also, the fastest conventional trains included in the study have a maximum speed of 260 km/h, whereas project speeds of 320 or 350 km/h are common today. Let's note that van [START_REF] Van Wee | Environmental impacts of high-speed rail links in cost-benefit analyses: a case study of the dutch zuider zee line[END_REF] did include the Maglev train, though this is not a conventional model. In conclusion, HSL offers a specific domain with its own constraints, and previously published works lack the precision required to distinguish different variants designed with the same train technology.

The method proposed in our paper has been developed to assess alternatives during the pre-project (or design) phase of a railway project. Moreover, our method takes into account both the infrastructure and its operations phase. The discrete approaches normally proposed for LCA products are adopted herein and combined with vehicle dynamic models. Both these approaches offer a well-to-wheel perspective so as to be comprehensive and take into account all phases (construction, maintenance, operations).

In the following sections, the acronym PEAM (Project Energy Assessment Method) will be used to describe our particular model. This paper continues with a full description of PEAM, which will then be applied to a real case study, i.e. a European HSL project whose life cycle phases (construction, maintenance, operations) will be studied separately as regards energy consumption. The final section will discuss the applicability of PEAM to other impacts and other projects, along with its limitations.

The PEAM method for assessing project variants

Purpose of a variants assessment using LCA

At the project level of a railway, the various routes (i.e. current route and variants) are examined simultaneously (Figure 1). Application of the proposed model would thus contribute to the decision-making process by focusing on the main design parameters to obtain an objective criterion for choosing a variant. Such a model could avoid controversy, such as that illustrated by [START_REF] Leheis | High-speed train planning in France: Lessons from the Mediterranean TGV-line[END_REF], and promotes better practices at the project level.

The alignment of an HSL requires mild slopes and gentle curvature. These low flexibilities of the horizontal and vertical alignment are the main causes of environmental issues as regards territorial integration. In comparison with motorways, the geometric constraints of railways require greater topographic adaptation. More specifically, a radius of plan curvature and maximum passable slope for an HSL are respectively 5 km and 3.5 %, vs. 0.6 km and 5 % for motorways [START_REF] Setra | ICTAAL -instruction sur les conditions techniques d'amÉnagement des autoroutes de liaison[END_REF]RFF and SNCF, 2006). It is essential to minimize the impacts of these projects on the environment. The environmental impact of major infrastructure projects is not negligible, as demonstrated by [START_REF] Capony | Monitoring and environmental modeling of earthwork impacts: A road construction case study[END_REF], who compared earthworks impacts to traffic for a road project (25 km). The expected influential project parameters are: slope, radius of curvature, and length. An analysis of targeted objectives with an HS line model, as performed in [START_REF] Vandanjon | Practical guidelines for Life Cycle Assessment applied to railways project[END_REF], indicates that a continuous analysis of the HSL is indeed required in order to obtain a pertinent assessment. As for the structural assessment, some LCA studies focus on the railway track to generate environmental product declarations. A literature review has also been undertaken to highlight generic data at the infrastructure scale that may be reused in this context to help build our method. Botniabanan-AB (2010) expressed the functional unit as « 1 Ton.km », [START_REF] Lee | Comparison of environmental loads with rail track systems using simplified life cycle assessment (LCA)[END_REF] used « 1 km of track », and [START_REF] Rozycki | Ecology profile of the german high-speed rail passenger transport system, ice[END_REF] considered « 100 person.km » (although this was dedicated to a comparison between ballast track and concrete track). The type of railway i.e. high-speed, intercity and urban, was investigated by [START_REF] Chester | Life-cycle assessment of high-speed rail: the case of California[END_REF] and [START_REF] Federici | Air versus terrestrial transport modalities: An energy and environmental comparison[END_REF] in considering the following functional units: « Vehicle.km », « Passenger.km », and « Ton.km ». These authors focused on intermodal comparisons. Throughout all these studies therefore, it seems that the purpose of studying a given itinerary is not necessarily a typical goal of the LCA literature, although some of the individual line studies, e.g. [START_REF] Chang | Life cycle greenhouse gas assessment of infrastructure construction for california's high-speed rail system[END_REF], did assess greenhouse gas emissions for high-speed railway infrastructure construction in California. Cross-technology comparisons have been performed, as have comparisons across modes of transportation; however, the actual eco-design of a railway project has yet to be undertaken. In addition, other works have not taken into account all energy costs throughout the life of the project.

Model set-up 2.2.1. Goal and scope

The method developed herein is called PEAM; it provides an integrated framework for conducting energy assessments. PEAM includes a systemic approach based on decomposition into several functional units, including construction, maintenance, and operations of an HSL. As is typically proposed for output, each functional unit takes into account the processes of materials extraction, production, transportation (truck, train, boat), and on-site works using non-road equipment, with verification of material consumption. The main goal of this method is to estimate the energy consumed over the entire life cycle in adapting the LCA framework.

The models introduced to evaluate project energy consumption as regards construction and maintenance use data associated with a long list of materials and processes needed to build the linear portions of the route, along with the bridges, tunnels and viaducts. Rehabilitation of the road network after insertion of an HSL is also considered. The generic data were collected from the literature, whereas the specific project data were obtained from HSL bids, namely French projects for the present study. Such data collection methods and principles have also been discussed by [START_REF] Chang | Life cycle greenhouse gas assessment of infrastructure construction for california's high-speed rail system[END_REF] and [START_REF] Rozycki | Ecology profile of the german high-speed rail passenger transport system, ice[END_REF].

Our method is thus based on a continuous approach along the HSL alignment by taking into account these functional units (FU) and aggregating them to describe the entire HSL. Depending on the line design, some of these elementary FU, which turn out to be geometry-dependent, vary and therefore allow proceeding with a generic project approach. As for the other elementary FU, they can also be considered as generic and hence provide an energy consumption ratio per km of line. PEAM also assumes, as shown in Figure 1, that the traffic is being handled by several existing railway lines within a given territory. The method thus assesses different pre-project possibilities: the implementation of either one single new line or several lines.

PEAM proposes a combined approach for pre-project variant assessment that relies on a system dedicated to a flow of trains crossing a territory on an infrastructure with given longitudinal and transverse profiles. The whole environmental system can then provide mobility services within a territory for given number of years. As regards territorial aspects, the PEAM method includes localization concepts not usually considered within the LCA framework, which nonetheless offer a scalar assessment derived from the LCA in combination with vehicle dynamic modeling methods. This set-up takes into account the longitudinal track profile at the project scale and therefore the final HSL geometry sought by the project owner. The entire method, as presented herein, has been generated from previous work dedicated to project data collection: i) [START_REF] Vandanjon | Practical guidelines for Life Cycle Assessment applied to railways project[END_REF] concentrated on the purpose of a single HSL assessment using LCA at the project stage. Their work has been adapted here to make use of the available data, albeit at the pre-project stage, which has lead to introducing the longitudinal profile of variants as considered through variant lengths, slopes and curves;

ii) [START_REF] Bosquet | Model of high-speed train energy consumption[END_REF] proposed a complete train model, as validated at the full-scale by tests using high-speed trains. The applicability of their method will be investigated in this paper.

Itinerary discretization and specific functional units

PEAM is intended to offer a unified LCA-based method for both construction/maintenance assessment and operations modeling, whereas these two topics are usually investigated separately in the literature. Any itinerary is naturally tied to a given high-speed line project, since it takes into account local territorial constraints in defining the numbers of bridges, tunnels, viaducts and earthmoving equipment (due to the route alignment) as well as the distances between the points to be connected in crossing the given territory (see Fig. 1, points A1 and B1). Each project is thus different and a case-by-case assessment needs to be performed. PEAM however defines a global functional unit to assess any itinerary of interest, as displayed in Figure 2, which has been broken down into 3 subfigures: Figs. 2a, 2b and 2c. Figure 2a shows the complete functions associated with the full HSL. The total project length is recorded as l tot . This total function, associated with a total functional unit, is then subdivided into 2 subunits: i) Fig. 2b pertains to the use phase; ii) Fig. 2c pertains to the construction and maintenance of linear infrastructure sections and works, as well as to the consequences of line integration into the studied territory (new construction). The infrastructure phase is broken down into 6 elementary functional units (efu 1 through 6), as described below.

Hence, some elementary functional units can be considered as either generic or specific, as will be discussed below:

• efu 1: This efu assesses the total earthworks involved in the resulting cut and fill on the project area using local data on the new project. For the works assessments component, consumption ratios are used. [START_REF] Capony | Monitoring and environmental modeling of earthwork impacts: A road construction case study[END_REF] applied such ratios to cut volumes, fill volumes, on-site transport distances, earthmoving distances (cuts to fills) and materials supplied from outside the HSL works area. This efu uses specific data at the pre-project step;

• efu 2: This efu assesses construction of the civil engineering structures including bridges, viaducts and tunnels.

The tunnel length (l t , Fig. 2c), viaduct length (l v , Fig. 2c) and bridge volume (v b , Fig. 2c) are calculated from this new project layout. Beginning with the unitary result expressed per kilometer and cubic meter for each kind of structure, the energy can then be calculated. This efu is project-specific;

• efu 3: This efu assesses the total number of km of sewer drain construction. l tot (Fig. 2c) is used for this purpose. This efu can be considered as generic for French HSL;

• efu 4: This efu assesses total road rehabilitation and construction based on a per-kilometer ratio along the total line length (l tot , Fig. 2c). This efu provides a generic assessment of road connections within France; • efu 5: This efu assesses the number of km of railway foundation construction and maintenance of railroad lines, ballast and sleepers (along the total length, l tot ). This efu is generic for railway projects in France;

• efu 6: This efu assesses the number of km of railway equipment construction (signaling, cables, catenary, etc.) along the total length, l tot . This efu is also generic for railway projects in France;

• efu 7: This efu assesses the operations phase (train traffic). It is specific to a given project and typically depends on the territory being crossed; hence, a model is presented to assess the line along its longitudinal profile..

Modeling steps

Since no studies in the literature that perform LCA actually deal with itineraries given that they focus on passenger mobility, the present study starts with a dedicated data collection effort specific to railway itinerary, as defined by the owner in calls for tender and contractor selection procedures. All documents provided by the French project owner (RFF) and made available for two high-speed lines already in service were then reviewed in order to analyze existing HSL projects.

To develop the first six efu, several case studies of high-speed railway lines were taken into account for data collection purposes. 400 km of existing lines were investigated as a basis for our assessment. From these projects, a set of generic life cycle inventory (LCI) data was extracted to assess the functional unit of a new project. The links between geometry and environmental systems considered are shown on Figure 3. This step led to considering elementary data at the efu i level, as expressed per kilometer of HSL.

Moreover, the power consumption of train motion (efu 7) is estimated in several steps, as indicated on Figure 4. The first step simulates the mechanical power provided by motors. Train characteristics, the track profile, travel time and initial speed constitute the required initial input. The second step outputs the estimated power consumed at the pantograph. Traction system losses need to be modeled, and the energy exchanged by the train is determined by

Energy production

French mix for French data -International mix for other data

Transport

Kilometer

Materials

Mass, volume

Equipment

Liter fuel, MWh

Track profile

Length Volume

Primary energy

Figure 3: System for performing construction/maintenance assessments.

mathematically integrating the power. Lastly, line losses are taken into account and the primary energy required to produce electricity is estimated.

Energy consumption modeling during the construction and maintenance phases

Each elementary functional unit (efu) is described by a system (see Fig. 3) composed of processes, input material flows and an output primary energy flow. The transport component is always a parameterized data element. In order to assess the environmental impact of an HSL itinerary, the functional unit must describe the technical features of railway track and traffic patterns along the itinerary as input into the track design.

The models applied to assess energy consumption for construction and maintenance use unit consumption data associated with an extensive list of materials tied to the processes required to build the linear portions of the line, along with the bridges, tunnels and viaducts. Such data, when in generic form, were collected from the literature and have been detailed in the present paper, whereas the specific data were obtained from HSL project bids, namely from French projects cited in the present study. This method and data collection principles were also employed by [START_REF] Chang | Life cycle greenhouse gas assessment of infrastructure construction for california's high-speed rail system[END_REF], who assessed greenhouse gas emissions for high-speed railway infrastructure construction in California.

All of the documents available for two French high-speed lines (LGV Est and LGV Rhin-Rhône) in eastern France were then reviewed in order to analyze existing projects. This protocol provided a description of material supply along a given project alignment, plus the generic data for the elementary functional units efu 2 through efu 6. It can be stated that efu 1 and efu 7 are specific to the route relative to the HSL case study and territory being crossed; moreover, these two units are based on the actual longitudinal profile. efu 2 to 6 are, on the other hand, correlated with the project length while remaining constant along the given line. Tables 1 and2 provide the details of material consumption for the studied project, as well as data sources for the energy consumption and energy mix.

The models for assessing energy consumption during construction and maintenance also use consumption data in association with an extensive list of materials and processes required to build the linear portions of the line, along with the bridges, tunnels and viaducts. Such data in generic form were collected from the literature and detailed herein, and once again the specific data were all obtained from HSL project bids, i.e. the cited French projects. As mentioned above, this method and data collection principles were displayed by [START_REF] Chang | Life cycle greenhouse gas assessment of infrastructure construction for california's high-speed rail system[END_REF] during their greenhouse gas emissions assessment for high-speed railway infrastructure projects in California.

Table 1 lists the descriptions of efu 1 and efu 2 in presenting the raw data tied to material consumption and transport. Table 2 then details these same data but for efu 3 through efu 6.

Modeling of energy consumption during the operations phase

In this section, the efu 7 evaluation will be completely described with respect to this paper's objective, namely providing a fine-tuned comparison of HSL variants in order to assess the project's energy consumption throughout its entire life cycle.

Dynamic model. This model development process began with a review of train consumption models. To estimate the electricity consumption of high-speed trains, the forces acting on the train were first investigated in the literature. [START_REF] Lukaszewicz | Running resistance -results and analysis of full-scale tests with passenger and freight trains in sweden[END_REF] and [START_REF] Rochard | A review of methods to measure and calculate train resistances[END_REF] offered interesting general formulations of operational performance as a function of train characteristics like mass, number of bogies, inter-vehicle gap and number of pantographs. To estimate energy consumption, the train is considered as a point with mass M [START_REF] Rochard | A review of methods to measure and calculate train resistances[END_REF]. Newton's second law is then applied at this point, as expressed in Equation 1. The total force conveyed to the drive wheels by the electric motor is computed in Equation 4.

As discussed in the introduction, the results of these publications are not directly applicable to this study because they lack the precision needed to distinguish two variants. For this reason, we relied on the methodology described in these papers to build a suitable model for our intended purpose.

The forces balance is obtained from Newton's second law applied to the train, i.e.:

k • M • γ = F -M • g • sin(α) -F r -F c : (1)
where on the left-hand side of Equation ( 1):

• M is the train mass;

• k is the coefficient of inertia of rotating masses (rather than modeling the entire mechanical system, it has been preferred to multiply the mass by a constant);

• γ is the longitudinal acceleration;

For the right-hand side of Equation (1):

• F denotes the total force conveyed to the drive wheels by the electric motor;

• g is the gravity acceleration and α is the local gradient of the line;

• F r is the resistance force;

• F c is the resistance force in curvature.

F r is composed of the following physical effects: i) rolling resistance, which is related to the contact wheel rail. As an initial approximation, it is considered to be constant (due to the sticking effect, this value is not the same when the train is stopped or set into motion); ii) mechanical resistance, which consists of a viscous friction Fc, mainly dependent on velocity, and dry friction F s, considered as constant (unless the train is starting up, like for the rolling resistance); and iii) aerodynamic resistance, related to drag coefficient Cx and weather conditions (e.g. wind, rain). This composite resistance depends primarily on the velocity squared. By taking the previous physical interpretation into account, this resistance force (F r ) is approximated by a second-order polynomial [START_REF] Rochard | A review of methods to measure and calculate train resistances[END_REF]:

F r = A + B • V + C • V 2 (2)
Where:

• V is the train velocity;

• A, B, C are coefficients dependent on the rolling stock.

F c (the resistance force in curvature) is modeled by using the classical formula given in both [START_REF] Fayet | Modélisation des réseaux électriques ferroviaires alimentés en courant alternatif[END_REF] and [START_REF] Rochard | A review of methods to measure and calculate train resistances[END_REF]:

F c = M • 9.81 • sin 0.8 R c (3)
• R c is the curvature radius in a horizontal plane.

To model driving, the driving rules need to be described. Two speed limits are introduced. The first limit is the set point speed. If the train speed lies below this set point (i.e. case 1), then F is positive. The second limit is the maximum speed. If train speed lies between the set point and the maximum limit (case 2), then F is zero. When the speed exceeds the maximum (case 3), F is negative. These three cases are summarized in Figure 5 with an example of set point and maximum speed limits. Driver behavior in cases 1 and 3 is modeled by means of a proportional integral corrector. The two coefficients are chosen so that force F is maximized if the speed deviation exceeds 10 m/s. The set point speed is chosen to satisfy the travel time constraint while the maximum speed limit is set by safety restrictions.

A classical numerical method (Adams method) is introduced to solve the differential equation. The initial conditions (position and speed) are selected on the basis of the study area. The integration of these ordinary differential equations yields the position, speed and force F provided by the motors along the section length. To estimate the mechanical power generated by the motor, F is multiplied by the speed:

P provided = F • V (4)
Figure 6 summarizes the steps required to obtain mechanical power. Energy pantograph simulation. As demonstrated by [START_REF] Jeunesse | La motorisation du TGV POS[END_REF], [START_REF] Lindgreen | Simulation of energy sinsumption and emissions from rail traffic[END_REF] and [START_REF] Boullanger | Modeling and simulation of future railways[END_REF], the electricity consumption can be deduced by introducing a ratio η that illustrates the efficiency of the traction system and its auxiliary equipment:

if P provided > 0 then

P = P provided • (1/η) + β if P provided < 0 then P = P provided • η + β
where :

• β is the auxiliary. It represents the power consumed to enhance comfort (heating, lighting, etc.) as well as for equipment, such as ventilation and cooling of propulsion equipment and compressed air supply for brakes;

• η is the traction system efficiency. As an initial approximation, this efficiency is considered constant.

The consumed power is then integrated in order to obtain energy consumption.

E = time P consumed (5)
Equation 5 implies that negative energy (i.e. when the train uses its regenerative brakes) is directly subtracted from the consumed energy, which is a key point in the energy balance of a high-speed train in comparison with other modes of transport like cars and airplanes.

Primary energy expended. The final model step is to estimate the primary energy consumed to supply the train.

Losses stem from two sources; i) the system for transporting electric power from the catenary to the power station; and ii) electric energy production, which generally converts thermal energy (nuclear, gas, etc.), hydro, solar or mechanical power into electric energy. The first category of losses depends on the transport system (voltage, frequency, etc.), whereas the second depends on each country's energy supply mix. As is the case for construction and maintenance, if the energy is consumed in France, the French energy mix will be considered. Other scenarios can easily be investigated.

Results

Presentation of case study

The European high-speed network has grown considerably in recent years. As observed in Figure 7, this network still has some missing connections, especially between Montpellier and Perpignan (see the black circle in the figure), which has been chosen as our project case study. To a greater extent further south, the section of line between Perpignan and Figueras has been in service since 2013. Over the northern part of the line, RFF (railroad owner) is currently building a section between Nimes and Montpellier. With the completed Montpellier-Perpignan line integrated into the actual network, trains will therefore be continuously circulating at high speed between Paris and Madrid.

The proposed new high-speed line should meet a certain number of demands resulting from a fast-growing transport sector, namely: i) capacity shortage problems now as important an issue as speed; ii) the increased volumes of railway freight as an alternative to road transport; iii) expanded regional express train service in an area with high population density along the coast and with high traffic levels. Lastly, the new Montpellier-Perpignan line can be qualified, according to the above considerations, as a major rail project to cross Europe and will handle 124 freight trains a day (124 FT/d) and 42 passenger trains a day (42 PT/d).

High speed line: studied in operation under construction

Scenarios considered

Of the 155-km long project line, the studied line covers about 25 km (this length will vary to some extent depending on the variant). Over this section, two scenarios are proposed: a new high-speed line dedicated to passenger trains (NPL), and a new mixed line for passenger and freight trains (NML). In both cases, the existing line (a conventional line -CL) will be kept in service. PEAM is therefore applied to compare both variants. Figure 8 shows the traffic distribution between the new line (connecting points A1 and B1) and the existing line (point A2 to point B2) for the studied scenarios. For the NPL, the traffic equals 42 PT/d on the new line and 124 FT/d on the conventional line. As for the NML, the traffic is: 42 PT/d plus 43 FT/d on the new line, and 81 FT/d on the conventional line.

The maximum overall capacity of the two projects differs. More NPL trains can cross the area because the NML is able to combine fast and slow trains, with traffic forecasts however far from reaching saturation.

The longitudinal profiles corresponding to the lines of these two scenarios differ as well. The maximum gradient of the NPL is 35 mm/m while this figure is 12.5 mm/m for NML as a result of the more stringent restrictions placed on freight trains. Moreover, the relevant lengths are different. Between the same pair of departure and arrival points (A1 and B1 in Fig. 8), the length equals 22.3 km for NPL and 25.3 km for NML. The infrastructure components, according to Figure 2 (line, bridges, etc.), are also differently distributed for these two scenarios. The total length of tunnels, viaducts and bridges varies, and this variation influences the overall assessment of the functional unit. As a final consideration, the FU for both variants is: 42 PT plus 124 FT per day during a 50-year period within a given area spanning 25 km x 5 km.

Construction phase results

The energy consumption evaluation for the efu and variants is obtained from the data as follows. Data on efu 1 and efu 2 have been compiled in Table 3 as regards volumes moved or worked. For efu 1, i.e. earthworks data, the excess excavated materials are moved in assuming a 15-km transport distance. As for efu 2, several types of civil engineering structures are taken into account (bridges, viaducts, tunnels, trenches); moreover, cement CEM I is considered for each structure. According to the total energy consumption result, the NML (Fig. 9) costs more energy per FU: 3,300 Tera Joules (TJ) for NML vs. 1,800 TJ for NPL. As observed in Figure 9, which exhibits the ratio of each efu to the whole construction task, this finding is mainly due to the tunnels required for the mixed line (2,054 TJ for NML vs. 207 TJ). It is also found that the earthmoving phase accounts for 36 % with NPL. The characteristics of spoils and banks are obviously critical for this scenario.

Maintenance phase results

Maintenance works only concern efu 5 as regards the FU chosen for the present study. Maintenance operations under both scenarios are assumed to be equivalent per km in the present study since no discriminating data are available to provide any means of distinction. The difference in traffic between the two lines is not significantly sufficient to highlight any change in maintenance principle. Hence, the hypotheses forwarded are that the rail is renewed after 25 and 50 years, the ballast is renewed after 30 and 50 years, and sleepers after 50 years. Similar process, material and transport data to those given in Table 2 for construction were used in these calculations. The energy consumption taken into account includes materials production and installation. This part of the life cycle is given in Figure 10 vs. time for NPL.

Operations phase results

Conventional parameters were used to run the train model. The mass M, coefficient of rotating masses k, coefficients A, B, C and η were extracted from [START_REF] Jeunesse | La motorisation du TGV POS[END_REF], [START_REF] Andersson | Energy consumption and related air pollution for Scandinavian electric passenger trains[END_REF], [START_REF] Fayet | Modélisation des réseaux électriques ferroviaires alimentés en courant alternatif[END_REF] and [START_REF] Rochard | A review of methods to measure and calculate train resistances[END_REF]. They have already been validated as a comprehensive parameter set [START_REF] Bosquet | Model of high-speed train energy consumption[END_REF]. For electricity production, French data were used. The simulations were performed for a 50-year use period, as defined by the FU.

Results are shown in Figure 11. The consumption model can clearly differentiate the two scenarios. As observed, NML is more expensive from an energy point of view, i.e. with a 35 % higher price than NPL. This difference is mainly due to the fact that NPL is shorter than NML. Furthermore, for freight traffic, the conventional line is more price competitive than the mixed line with respect to energy. The NPL therefore is more attractive for freight traffic from a strict energy point of view.

Total life cycle results (efu 1 to efu 7)

The total primary energy results (operations, construction and maintenance) were calculated and have been displayed in Figure 12. Let's note from the graph that NML consumes more energy, i.e. 30 % after 50 years in comparison with NPL. This graph also compares the energy spent on infrastructure construction and (if applicable) maintenance with the primary energy spent during operations. After a 50-year service life, infrastructure construction and maintenance in the Montpellier-Perpignan case study account for 16 % of the total energy cost for NML and 20 % for NPL.

Discussion

New results at the pre-project stage with PEAM

The purpose of this paper has been to assess, at the pre-project stage, a new HSL project linking two points in a given territory and ensuring a given train traffic level over a 50-year service life. A conventional line (freight and regional trains) already exists within the study area, thus generating variants that include both a new line and a mixed line. A number of methodological considerations for the LCA of HSL projects have already been proposed [START_REF] Vandanjon | Practical guidelines for Life Cycle Assessment applied to railways project[END_REF]. The results obtained here provide the per-km primary energy consumption ratios for efu 3 through efu 6, which will prove useful for any subsequent project assessment. Moreover, the other territory-dependent efu (1 and 2) have been carefully examined in the present paper using available data, such as earth movement volumes and longitudinal profiles.

The PEAM method has been developed in collaboration with French railway owner RFF and applied at several steps of the HSL life cycle; it appears to offer a powerful tool for assessing projects during the preliminary stage, when defining variants within a given territory.

One major contribution of this study is to clarify the issues presented in public debate. In France, for this type of project, a public hearing is required. Such a hearing was held for the Montpellier-Perpignan HSL (recorded hearing minutes are available at http://cpdp.debatpublic.fr/cpdplnmp/). During the debates, some non-specialists expressed the opinion that a mixed line is better for the environment than an exclusively passenger line because it offers more opportunities for the modal shift of freight from road to rail. This perspective however did not take into account the additional energy cost associated with a mixed line. Another important contribution of our work has been to clarify debate arguments by providing an energy assessment methodology.

Applicability of PEAM to other projects

The PEAM method may be applied to projects of various sizes, i.e. lengths of between 10 km and 1,000 km. Since we are encouraging application of this methodology to other projects, let's caution that the figures found in our study cannot be directly extrapolated to another country. For example, the ratios obtained between the efu integrate the indirect impacts of energy sources that are typical of France (3.08).

Hypotheses and limits of PEAM

Moreover, even though many indicators can be calculated according to LCA, the present method has only been applied to studying energy consumption as one of the major parameters in any global assessment, which also introduces sustainability considerations. In the context of increasing energy prices per MJ sold on the market due to future depletion concerns, energy remains a very understandable parameter that can easily be considered on its own for discussion purposes and decision-making among stakeholders and users. Moreover, it can be easily measured and modeled by researchers under various methodological assumptions, including train consumption modeling on an itinerary and LCA applied to technical construction strategies.

The life cycle of vehicles has not been included in the present study since it remains constant in both scenarios.

In our method, only the service life has been investigated and the end of life has not been integrated. For the time being, only energy has been presented, and it was shown that our method yields different results between the two variants. For a fuller understanding of these scenarios, it would be beneficial to calculate other impacts classically found in LCA and then to verify the extent to which energy results and variant rankings remain similar for other indicators.

In examining these variants, road rehabilitation also induces mobility changes that are not addressed at this stage but could be integrated in order to conduct a more sustainable approach.

The time required to connect two points within a given territory differs for each variant. From a mobility point of view, travel time should be taken into account.

Although not yet integrated into our functional unit, future works should incorporate the notion of travel time.

As a final consideration, these results, as would be expected, are highly dependent on the hypotheses and questions posed. For all these reasons, it would be useful to continue developing this method in conjunction with other indicators so as to check for other important impacts to include in an eco-design approach.

Conclusion

The construction of a new railway line may be necessary for several reasons, among which increasing capacity, reducing travel time and allowing for other train gauges. The new infrastructure not only will have an impact on other modes of transport (road and air) but will also change the distribution of train traffic. The challenge for this paper has consisted of collating the results from previous models developed separately into an integrated approach for a new HSL project, with the ultimate aim of performing an eco-design at the project level in using energy consumption as a valuable criterion for resource conservation. To achieve this goal, numerical models were developed to simulate energy consumption during both operations and construction/maintenance. An efficient methodology called PEAM was proposed for comparing variants at the pre-project stage, as regards energy consumption. Construction, maintenance and operations phases can thus now be investigated comprehensively, and their respective influences determined using quantified energy results.

Figure 1 :

 1 Figure 1: Theoretical example of traffic flow through a territory.

Figure 2 :

 2 Figure 2: Line discretization for itinerary assessment and elementary functional units.

Figure 4 :

 4 Figure 4: Steps during the operations phase for model development.

Figure 5 :

 5 Figure 5: Driving model hypotheses.

Figure 6 :

 6 Figure 6: Dynamic model steps.

Figure 7 :

 7 Figure 7: European high-speed line (speed > 220 km/h) in 2009, RFF (2009), with the situation of the Montpellier-Perpignan project.

Figure 8 :

 8 Figure 8: Traffic as a function of variant scenario.

Figure 9 :

 9 Figure 9: Construction energy requirements (NML vs. NPL).

Figure 10 :

 10 Figure 10: Cumulative maintenance energy for NML.

Figure 11

 11 Figure 11: A comparison of energy consumed during the 10-year operations phase -efu 7.

Figure 12 :

 12 Figure 12: Cumulative energy spent during the operations and construction phases.

Table 1 :

 1 Description of efu 1 and efu 2.

						Distance
			Efu i	Main processes	build Efu i Amount to	(km) supply	Source
	Efu 1	Earthworks		Cuts (l/m 3 ) Fills (l/m 3 ) Rock cuttings (l/m 3 ) Crushing, screening (l/m 3 )	0.20 Depending 0.25 to 0.30 on the 0.35 case study 0.90 to 1.30	Jullien et al.2012
				Transport (l/m 3 /km)	0.40
			Bridges	Reinforcing steel (t/m 3 ) Concrete (m 3 /m 3 ) Equipement fuel (l/m 3 )	0.07 1.04 1.97		Process "Ecoinvent" adapted
				Reinforcing steel and
				structural steel (t/km)	6 800
			Viaducts	Concrete (m 3 /km) Aggregates (t/km) Hydraulically bound	23 500 32 000		Process "Ecoinvent" adapted
				materials (t/km)	4 200
	Efu 2	Civil engineering	Two-tube tunnel dug with TBM	Equipement fuel (l/m 3 ) Reinforcing steel (t/km) Concrete (m 3 /km) TBM (MWh/km) Equipement fuel (l/km)	290 000 2 100 47 500 7 850 20 000		Process "Ecoinvent" adapted Ecorce 2.0 AFNOR FDP 01-015 Ecorce 2.0
							Process "Ecoinvent"
			Two-tube tunnel	Reinforcing steel (t/km)	4 300		adapted
			dug conventionally	Concrete (m 3 /km) Equipement fuel (l/km)	70 000 500 000		Ecorce 2.0
							Process "Ecoinvent"
			Trench covered	Reinforcing steel (t/km)	8 800		adapted
			with two tubes	Concrete (m 3 /km) Equipement fuel (l/km)	88 000 450 000		Ecorce 2.0
							Process "Ecoinvent"
			Monotube	Reinforcing steel (t/km)	3 600		adapted
			tunnel dug conventionally	Concrete (m 3 /km) Equipement fuel (l/km)	58 000 260 000		Ecorce 2.0
							Process "Ecoinvent"
			Covered trench	Reinforcing steel (t/km)	5 800		adapted
			monotube	Concrete (m 3 /km) Equipement fuel (l/km)	58 000 420 000		Ecorce 2.0

Table 2 :

 2 Description of efu 3 through 6.

					Distance
		Efu i	Main processes	build Efu i Amount to	(km) supply	Source
			PVC (t/km)	69	100 Process "ecoinvent" adapted
	Efu 3	Drainage	Reinforcing steel (t/km) Aggregates 1 (t/km) Cement (t/km) Concrete (t/km)	2.4 245 47 316	30 300 100	Ecorce 2.0
			Equipment fuel (l/km)	2 400
			PVC (t/km)	0.14	200	Process "ecoinvent"
	Efu 4	Road rehabilitation	Reinforcing steel (t/km) Aggregates 2 (t/km) Cement (t/km) Concrete (t/km) Bitumen (t/km) Hydraulically bound materials (t/km)	16 5 900 88 41 47 3 100	30 300 30 270 100	adapted Ecorce 2.0
			Asphalt concrete (t/km)	900	30
			Equipment fuel (l/km)	1 500
			Reinforcing steel
	Efu 5	Railway foundation	(railway sleeper) (t/km) Steel rail (t/km) Cement (railway sleeper) (t/km) Concrete (railway sleeper) (t/km) Ballast (t/km)	35 240 130 1 040 8 300	500 500 150 30 200	Process "ecoinvent" adapted
			Equipment fuel (l/km)	55 000
			Steel (t/km)	42		Ecorce 2.0
			Aluminium (t/km)	0.11
	Efu 6	Equipment	Bronze (t/km) Copper (t/km) Cast iron (t/km) Stainless steel (t/km) Concrete (t/km) HDPE (t/km) PVC (t/km)	0.01 5.2 4.5 0.003 179 4.5 1	Average 130	Process "ecoinvent" adapted Ecorce 2.0 Process "ecoinvent" adapted
			Aggregates (t/km) Cement (t/km)	290 1.7		Ecorce 2.0
			Optical fiber (t/km)	0.5		Process "ecoinvent" adapted
			Equipment fuel (l/km)	2 400		Ecorce 2.0

Table 3 :

 3 Refined data for efu 1 and efu 2 in relation to the variants.

	TOTAL	Mixed line NML	Passenger line NPL
		Length: 25.273 km	Length : 22.287 km
	Efu 1 / Cuts	1,700,000 m 3	10,000,000 m 3
	Efu 2 / Bridges	49,901 m 3	80,028 m 3
	Efu 2 / Viaducts	1.592 km	1.061 km
	Efu 2 / Tunnels (tunneling)	3.2+2.75 km	0
	Efu 2 / Tunnel (conven-	1.5 km	0.3 km
	tional)
	Efu 2 / Trenches (covered)	0.36 km	0.6 km

including concrete aggregates

including concrete aggregates, hydraulically bound material aggregates, and asphalt concrete aggregates

Acknowledgements

The study was conducted with the support of Réseau Ferré de France (RFF), owner of the national railway network, as part of a partnership with Ifsttar. The authors would also like to thank Inexia (contract manager) for providing data relative to a viaduct. Special gratitude is also addressed to: Olivier Cazier, Edouard Parant, Nicolas Aladern from RFF, and Amandine Fargier and Christophe Ropert from Ifsttar.


Agnès Lagnoux

Sabine Mercier

Pierre Vallois

Probability density function of the local score position

Keywords: Reflected Brownian motion, Brownian bridge, Brownian excursions, local score, sequence analysis

We calculate the probability density function of the local score position on complete excursions of a reflected Brownian motion. We use the trajectorial decomposition of the standard Brownian bridge to derive two different expressions of the density: the first one is based on a series and an integral while the second one is free off the series.

Introduction

This work is motivated by biological sequence analysis (e.g. DNA or proteins) largely developed since the 90's with the creation of databases. The local score is an usual tool to point out atypical segments of biological sequences. It has first been defined by Karlin and Altschul [START_REF] Karlin | Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes[END_REF]. Let (ε i ) i 1 be a sequence of i.i.d. random random variables, such that E[ε i ] = 0 and Var(ε i ) = 1. The random walk (S n ) n 1 associated with (ε i ) i 1 is:

S n := n i=1 ε i , n 1, S 0 = 0.
Email addresses: lagnoux@univ-tlse2.fr (Agnès Lagnoux), mercier@univ-tlse2.fr (Sabine Mercier), pierre.vallois@univ-lorraine.fr (Pierre Vallois)

The Lindley process (U n ) n 1 and the local score process (H n ) n 0 are respectively defined as:

U n := S n -min 0 k n S k , n 1, (1.1)
H n := max

0 k n U k = max 0 i j n (S j -S i ), n 0. (1.2)
This classical setting has been extended in many directions (see [START_REF] Karlin | Limit distributions of maximal segmental score among Markov-dependent partial sums[END_REF][START_REF] Mercier | Exact distribution for the local score of one i.i.d. random sequence[END_REF][START_REF] Hassenforder | Exact Distribution of the Local Score for Markovian Sequences[END_REF][START_REF] Mercier | A Hidden Markov Model for the Local Score of One Sequence[END_REF]).

For biological applications, the distribution of H n , when n is large, plays an important role. Here we do not study this question, the interested reader can consult [START_REF] Guedj | Detecting Local High-Scoring Segments: a First-Stage Approach for Genome-Wide Association Studies[END_REF][START_REF] Fariello | Accounting for Linkage Disequilibrium in genome scans for selection without individual genotypes : the local score approach[END_REF]. The Donsker theorem (see Section 2.10 in [START_REF] Billingsley | Convergence of probability measures[END_REF]) permits to define the local score in continuous time, see for instance Theorem 1 in [START_REF] Daudin | Asymptotic behavior of the local score of independent and identically distributed random sequences[END_REF] and [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF]. The underlying process is the standard Brownian motion (B(t); t 0). It can be easily proved (see [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF]) that the Lindley process (U k / √ n) 0 k n can be approximated by:

U (t) := B(t) -inf 0 s t B(t), 0 t 1. (1.3)
Recall that ( U (t); t 0) is distributed as the reflected Brownian motion (U (t); t 0) where U (t) := |B(t)|, t 0.

(1.4)

According to (1.1) -(1.4), we define the local score in continuous time as:

U (t) := sup 0 s t U (s), t 0. ( 1.5)
Let f (t) be the unique time which achieves the maximum of U over [0, t]: f (t) := sup{r t; U (r) = U (t)}, t 0 (1.6)

We say that the maximum U (t) occurs on a complete excursion if f (t) g(t), where g(t) := sup{s t; U (s) = 0}, t 0.

(1.7)

We have calculated in [START_REF] Lagnoux | Probability that the maximum of the reflected Brownian motion over a finite interval [0, t] is achieved by its last zero before t[END_REF] the probability of the event {f (t) g(t)}. Here we deal with the local score U * determined on complete excursions:

U * (t) := U (g(t)) = sup 0 s g(t)
U (s), t 0.

(1.8) f * (t) := sup{r g(t) ; U (r) = U * (t)}, t 0.

(1.9)

Now we are able to define the left end-point g * (t) of the excursion straddling f * (t)

g * (t) := g(f * (t)) = sup{r f * (t) ; U (r) = 0}, t 0.

(1.10)

In [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF], we have calculated the probability density function of (U * (t), f * (t)g * (t)). Unfortunately, it is complicated but the density function of U * (t) is rather simple since it equals the sum of an explicit series. We focus here on the distribution of g * (t). This random time can be interpreted in the setting of local score.

Recall that the scaling property of the Brownian motion implies that g * (t) is distributed as tg * (1). For that reason, we only consider in the sequel t = 1. For simplicity, denote g * (1) by g * . We have drawn a trajectory of (U (t) ; 0 t 1) (see Figure 1) and indicated the variables introduced above.

The main result and the scheme of its proof

Before stating the main result of the paper, let us introduce (i) The probability distribution function of g * is given by

h(x) := k 1 (-1) k+1 k cosh 2 (kx) . ( 2
p g * (y) = 1 2π y(1 -y) +∞ 0 ln 1 - π 2 (1 -y) 4ys h( √ s) √ s ds, 0 y 1.
(2.12)

(ii) The probability distribution function of g * is also given by

p g * (y) = 1 πy +∞ 0 ln |cot s| ds cosh 2 s 1-y y , 0 y 1. (2.13)
This new result is in the stream of previous papers [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF][START_REF] Lagnoux | Probability that the maximum of the reflected Brownian motion over a finite interval [0, t] is achieved by its last zero before t[END_REF][START_REF] Lagnoux | Statistical significance based on length and position of the local score in a model of i.i.d. sequences[END_REF]. Among the three studies, two are theoretical [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF][START_REF] Lagnoux | Probability that the maximum of the reflected Brownian motion over a finite interval [0, t] is achieved by its last zero before t[END_REF]. The last one is a review of asymptotic distributions of the local score in i.i.d. models and also contains illustrative simulations and statistical tests.

Let us explain the scheme of the proof of Theorem 2.1. Our approach is based on the identity (in distribution): Proposition 2.2. We have

g * (d) = g(1)g b (2.16)
where g(1) and g b are independent r.v. (2.17)

We easily deduce g(1)g

P (g(1) ∈ dx) = 1 π x(1 -x) ✶ [0,1] (x) dx.
(2.18)

Hence, it remains to determine the distribution of g b . One step in this direction comes from [START_REF] Pitman | Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude[END_REF]Theorem 2]. Before stating the result, let us fix notation. Let (L(t), t 0) be the local time process at 0 related to the Brownian motion (B(t), t 0) and (τ s , s 0) be its right inverse. Let us consider the r.v. ξ distributed as T 1 (R) with

T x (R) = inf{s 0 ; R(s) = x}, x > 0. (2.19) (R(s), s 0), introduced in [4]
, stands for a 3-dimensional Bessel process started at 0 and will be extensively used in the sequel. The density p ξ is explicitly known (see [START_REF] Biane | Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions[END_REF]) and is given by

p ξ (u) = 1 √ 2πu 3/2 k∈Z -1 + (1 + 2k) 2 u exp - (1 + 2k) 2 2u (2.20) = d du k∈Z (-1) k exp - k 2 π 2 u 2 , u > 0. (2.21)
We always assume in the sequel that ξ and (U (t)) t 0 are independent.

Proposition 2.3. Let f : R → R be a bounded Borel function. Then

E[f (g b )] = √ 2π 2 R 2 + E f l 2 τ 1 l 2 τ 1 + 4t 1 √ l 2 τ 1 + 4t ✶ {lU(τ1)<1} p ξ (t)dldt. (2.22)
The proof of Proposition 2.3 is postponed in Section 3.1. Integrating with respect to l in (2.22) and making the change of variable u = l 2 τ 1 /(l 2 τ 1 + 4t), considering that τ 1 and t are fixed, permits to obtain the density function of g b .

Corollary 2.4. [Probability distribution function of g b ]

The probability distribution function of g b is given by

p g b (u) = √ 2π 4 1 √ u(1 -u) +∞ 0 E 1 √ τ 1 ✶ {u(4tU (τ 1 ) 2 +τ 1 )<τ 1 } p ξ (t)dt✶ [0,1] (u

Proofs

We keep the notation introduced in Section 2.

3.1. Proof of Proposition 2.3 1) By definition, g b is defined via steps (1.7)-(1.10) where we substitute |b| for U . Theorem 2 [START_REF] Pitman | Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude[END_REF] is convenient for our purpose since it expresses the law of (b(s); 0 s 1) conditionally on b * , with two independent pieces of Brownian trajectories. Let us introduce br * (u) =

1 √ σ 1 +σ 1 B (u(σ 1 + σ1 )) , 0 u σ 1 σ 1 +σ 1 1 √ σ 1 +σ 1 B ((1 -u)(σ 1 + σ1 )) , σ 1 σ 1 +σ 1 u 1 (3.24)
where B and B are two independent standard Brownian motions and σ 1 = inf{t 0; B(t) = 1} and σ1 = inf{t 0; B(t) = 1}.

(br * (u); 0 u 1) is the concatenation of (B(u); 0 u σ 1 ) and ( B(t); 0 t σ1 ) with the scaling in space (respectively time) 1/ √ σ 1 + σ1 (resp. 1/(σ 1 + σ1 )). Lemma 3.1. [Theorem 2 in [START_REF] Pitman | Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude[END_REF]] For every non-negative measurable function F defined on the path space C([0, 1]),

E[f (g b )] = 2 √ 2πE f g(σ 1 ) σ 1 + σ1 1 √ σ 1 + σ1 ✶ {B(σ1)∧ B(σ 1 )>-1} (
E[f (g b )] = 2E[f (g b )✶ {b(1)>b(1)} ] = 2 √ 2πE    f (ρ) √ σ 1 + σ1 ✶    sup 0 u 1 br * (u) > -inf 0 u 1 br * (u)       ,
where ρ is the starting point of the highest excursion of (br * (s); 0 1). See Figure 3 for more precisions. We remark that (σ 1 + σ1 )ρ = g(σ 1 ) (see Figure 4) and sup

0 u 1 br * (u) > -inf 0 u 1 br * (u) = inf 0 u σ 1 +σ 1 X(u) > -1 = inf 0 u σ 1 B(u) ∧ inf 0 u σ1 B(u) > -1 .
Identity (3.27) follows immediately.

2) Remark that the expectation in the right hand side of (3.27) involves the law of (3.28)

First we determine the law of σ1 , B(σ 1 ) , see Lemma 3.3 below. Second we consider more generally the law of (g(σ 1 ), σ 1 , B(σ 1 )), see Lemma 3.5.

Lemma 3.3. Let a > 0 and ϕ : [0, +∞[→ [0, +∞[. Then E[ϕ(σ 1 )✶ {B(σ 1 )>-a} ] = +∞ 0 ϕ(t)ss t (a, a + 1)dt where ss t (x, z) = 1 √ 2π k∈Z z -x + 2kz t 3/2 exp{-(z -x + 2kz) 2 /(2t)}. (3.29) Proof of Lemma 3.3 We have E[ϕ(σ 1 )✶ {B(σ 1 )>-a} ] = E[ϕ(σ 1 )|B(σ 1 ) > -a]P(B(σ 1 ) > -a).
By definition, P(B(σ

1 ) > -a) = P(inf t∈[0,σ 1 ] B(t) > -a) = P(σ 1 < σ -a ) = a/(a + 1)
and conditionally on {B(σ 1 ) > -a}, (a + B(t), 0 t σ 1 )

(d) = (R a (t), 0 t H a+1 ) where (R a (t), t 0) is a 3-dimensional Bessel process started at a and H z = inf{t 0, R a (t) = z}.
Assume that a < z. By Formula 2.0.2 p63 in [START_REF] Borodin | Handbook of Brownian motionfacts and formulae,Probability and its Applications[END_REF],

P(H z ∈ dt) = z a ss t (a, z)dt. ( 3
E[f (g b )] = 2 √ 2π +∞ 0 E ✶ {B(σ 1 )>-1} f g(σ 1 ) σ 1 + t 1 √ σ 1 + t ss t (1, 2)dt. (3.31)
Now we deal with the distribution of (g(σ 1 ), σ 1 ) conditionally on {B(σ 1 ) > -1}.

Lemma 3.5. Let ϕ be a bounded Borel function. Then,

E[ϕ(g(σ 1 ), σ 1 )✶ {B(σ 1 )>-1} ] = 1 2 R 2 + E ϕ(l 2 τ 1 , l 2 τ 1 + t)✶ {lU(τ1)<1} p ξ (t)dldt (3.32)
Proof of Lemma 3.5 From Proposition 4 in [START_REF] Vallois | Une extension des théorèmes de Ray et Knight sur les temps locaux Browniens[END_REF], σ 1g(σ 1 ) is independent of (g(σ 1 ), B σ 1 ) and distributed as ξ (see Section 2). Moreover,

B σ 1 = B g(σ 1 )
. Hence

E[ϕ(g(σ 1 ), σ 1 )✶ {B(σ 1 )>-1} ] = +∞ 0 E[ϕ(g(σ 1 ), g(σ 1 ) + t)✶ {B(σ 1 )>-1} ]p ξ (t)dt.
Recall that (L(t); t 0) is the local time at 0 of (B(t); t 0). Using several times Proposition 4 in [START_REF] Vallois | Une extension des théorèmes de Ray et Knight sur les temps locaux Browniens[END_REF] we get:

E[ϕ(g(σ 1 ), σ 1 )✶ {B(σ 1 )>-1} ] = +∞ 0 p ξ (t) +∞ 0 1 2 e -l/2 E[ϕ(g(σ 1 ), g(σ 1 ) + t)✶ {B(σ 1 )>-1} |L(σ 1 ) = l]dldt = 1 2 R 2 + E[ϕ(τ l , τ l + t)✶ {B(τ l )>-1} |B(τ l ) < 1]P(B(τ l ) < 1)p ξ (t)dldt = 1 2 R 2 + E[ϕ(τ l , τ l + t)✶ {B(τl)>-1,B(τl)<1} ]p ξ (t)dldt.
Using U = |B| and the scaling identity (U (τ l ), τ l )

(d)
= (lU (τ 1 ), l 2 τ 1 ) (see Formula (4.2) in [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF]) ends the proof of Lemma 3.5.

Remark 3.6. The law of (τ 1 , U (τ 1 )) is rather complicated, see Proposition 2.2 in [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF]. Lemma 3.5 does not permit to have easily an explicit formula of the density function of g(σ 1 ) conditionally on {B σ 1 > -1} and therefore to recover Lemma 3.3. Now we are able to end the proof of Proposition 2.3. Using Corollary 3.4 and Lemma 3.5, one gets

E[f (g b )] = √ 2π R 3 + E f l 2 τ 1 l 2 τ 1 + t 1 + t 2 1 √ l 2 τ 1 + t 1 + t 2 ✶ {lU(τ1)<1} • p ξ (t 1 )ss t 2 (1, 2)dldt 1 dt 2 = √ 2π 2 R 2 + E f l 2 τ 1 l 2 τ 1 + 4t 1 √ l 2 τ 1 + 4t ✶ {lU(τ1)<1} p ξ (t)dldt because p ξ * 2ss • (1, 2) is the probability distribution function of H 2 when R 0 = 0 and H 2 (d) = 4H 1 (d)
= 4ξ by the scaling property of the Bessel process.

Proof of (2.12) in Theorem 2.1 3.2.1. A first formula of the probability distribution function of g *

Our starting point is obviously Proposition 2.2. Note that the density function of g( 1) is explicit and given by (2.18) and (2.23) shows that the one of g b is expressed via the unknown quantity:

I(u) := R + E 1 √ τ 1 ✶ {u(4tU(τ1) 2 +τ 1 )<τ 1} p ξ (t)dt, 0 < u < 1.
(3.33)

The main result of this subsection is Corollary 3.9.

Lemma 3.7. For any 0 < y < 1, one has

p g * (y) = - 1 2 √ 2π 1 √ y lim ε→0 1-ε y s y du (1 -u) u(u -y) dI(s). ( 3
E[f (g * )] = 1 π 1 0 E[f (xg b )] x(1 -x) dx = 1 2 √ 2π 1 0 f (y) y(u -y) I(u) √ u(1 -u) ✶ {0 y u} dydu (3.35)
where the last equality comes from the change of variable ux = y for a fixed u. Hence, using that I(1) = 0 and I is decreasing:

p g * (y) = 1 2 √ 2π 1 √ y 1 y I(u) u(u -y)(1 -u) du = - 1 2 √ 2π 1 √ y 1 y 1 u(u -y)(1 -u) 1 u dI(s) du = - 1 2 √ 2π 1 √ y lim ε→0 1-ε y s y du u(u -y)(1 -u) dI(s).
As shows (3.35) there is a singularity at u = 1. This explains why we introduce the cutoff ε. Now our strategy is to compute I (Lemma 3.8) and then its derivative. This leads to Corollary 3.9, which is the first main step of the proof of (2.12).

Lemma 3.8. We get

I(u) = 2 π + 2 k 1 (-1) k I 1 k 2 π 2 8 1 -u u , 0 < u < 1 (3.36)
where

I 1 (v) := 1 √ 2π +∞ 0 1 √ s 1 cosh 2 ( √ s + 2v) ds. (3.37)
Proof of Lemma 3.8 1) Let h 1 be the real-valued function defined by

h 1 (u) := 2 k 1 (-1) k exp - k 2 π 2 u 2 , u > 0. (3.38) By (2.21), p ξ = h ′ 1 , then for any t 1 > 0, t 1 0 p ξ (t)dt = +∞ 0 p ξ (t)dt - +∞ t 1 h ′ 1 (t)dt = 1 + h 1 (t 1 ). Since τ 1 (d)
= 1/N 2 with N ∼ N (0, 1), the above identity and (3.38) imply:

I(u) = E 1 √ τ 1 1 + h 1 1 -u u τ 1 4U (τ 1 ) 2 = 2 π + 2 k 1 (-1) k I 1 k 2 π 2 8 1 -u u
where

I 1 (v) = E 1 √ τ 1 exp -v τ 1 U (τ 1 ) 2 , v > 0. (3.39)
2) We calculate I 1 . Using the identity

+∞ 0 1 √ t e -ta dt = π a ,
with a = τ 1 and inverting and E in (3.39) leads to:

I 1 (v) = 1 √ π +∞ 0 E exp -τ 1 t + v U (τ 1 ) 2 dt √ t = 1 √ π R 2 + E exp -τ 1 t + v x 2 |U (τ 1 ) = x 1 x 2 e -1/x dx dt √ t , since 1/U (τ 1
) is exponentially distributed (see [START_REF] Vallois | Sur la loi conjointe du maximum et de l'inverse du temps local du mouvement brownien: application à un théorème de Knight, Stochastics Stochastics Rep[END_REF]Theorem 1]). We claim:

E e -µτ 1 |U (τ 1 ) = x e -1/x x 2 = 2µ sinh 2 (x √ 2µ) exp -2µ coth(x 2µ) , µ > 0.
(3.40) Indeed, by Proposition 2.2 and Formula (4.13) in [START_REF] Chabriac | Elements related to the largest complete excursion of a reflected Brownian motion stopped at a fixed time. Application to local score[END_REF],

E e -µτ 1 ✶ {U (τ 1 )<x} = exp -2µ coth(x 2µ) (3.41) = x 0 E e -µτ 1 |U (τ 1 ) = y 1 y 2 e -1/y dy, (3.42) (3.43)
and a derivative with respect to x leads to (3.40).

Using (3.40) and the change of variable u = 2 (tx 2 + v) for a fixed x,

I 1 (v) = 1 √ π R 2 + 2(tx 2 + v) sinh 2 2(tx 2 + v) • exp -2 t + v x 2 coth 2(tx 2 + v) dxdt x 2 √ t = 1 2 √ π R 2 + u x 3 u 2 -v sinh 2 ( √ u) exp - √ u x coth( √ u) ✶ {u>2v} dudx = 1 2 √ π +∞ 2v 1 u 2 -v cosh 2 ( √ u) +∞ 0 we -w dw du = 1 √ 2π +∞ 0 1 √ s ds cosh 2 ( √ s + 2v)
where we make the change of variable w = √ u coth( √ u)/x for a fixed u.

Corollary 3.9. We have

p g * (y) = π 8 √ y lim ε→0 k 1 (-1) k+1 k 2 δ k (y, ε), 0 < y < 1, (3.44)
where

δ k (y, ε) := +∞ 0 1 √ v 1-ε y 1 s 2 1 v + k 2 π 2 4 1-s s sinh cosh 3 v + k 2 π 2 4 1 -s s × s y 1 (1 -u) u(u -y) du dsdv. (3.45)
Proof of Corollary 3.9 We take the derivative in (3.36) and (3.37) and we get

I ′ (s) = - π 2 4s 2 k 1 (-1) k k 2 I ′ 1 k 2 π 2 8 1 -s s = π 2 4s 2 2 π k 1 (-1) k k 2 • +∞ 0 1 v v + k 2 π 2 4 1-s s sinh cosh 3 v + k 2 π 2 4 1 -s s dv (3.46)
that plugged into (3.34) leads to the required result.

3.2.2.

A second formula of the probability distribution function of g * From (3.45), we see that δ k (y, ε) is a triple integral. We reduce it to a simple integral see Lemma 3.10 below. Then (3.44) implies that p g * (y) is a limit of a single integral (see Corollary 3.11). The limit can be calculated, cf. Lemma 3.12, that proves (3.44).

Lemma 3.10.

Letting ε ′ = ε/1 -ε, one has δ k (y, ε) = 4 k 2 π 2 1 √ 1 -y +∞ 0   ρ 1 (ε ′ ) √ s - ρ 2 (s, ε ′ ) s + π 2 4 ε ′   kds cosh 2 k s + π 2 4 ε ′ (3.47)
where

ρ 1 (ε ′ ) := ln √ 1-y+ √ 1-y(1+ε ′ ) √ 1-y- √ 1-y(1+ε ′ ) and ρ 2 (s, ε ′ ) := ln u 1 (s,ε ′ )+u 2 (s,ε ′ ) |u 1 (s,ε ′ )-u 2 (s,ε ′ )| (3.48) with u 1 (s, ε ′ ) := 4s π 2 (1 -y -yε ′ ) and u 2 (s, ε ′ ) := 4s π 2 + ε ′ 1 -y . (3.49)
Proof of Lemma 3.10 In (3.45), we directly integrate with respect to s and then we make the change of variable w = (1u)/u:

δ k (y, ε) = 4 k 2 π 2 R 2 + ✶ {y u 1-ε} (1 -u) uv(u -y) •     1 cosh 2 v + k 2 π 2 4 ε ′ - 1 cosh 2 v + k 2 π 2 4 1-u u     dudv = 4 k 2 π 2 1-y y ε ′ dw w 1 -y(1 + w) • R +     1 cosh 2 v + k 2 π 2 4 ε ′ - 1 cosh 2 v + k 2 π 2 4 w     dv √ v = 4 k 2 π 2 (q 1 -q 2 )
where

q 1 := 1-y y ε ′ dw w 1 -y(1 + w)     +∞ 0 dv √ v cosh 2 v + k 2 π 2 4 ε ′     q 2 := 1-y y ε ′ dw w 1 -y(1 + w)   +∞ k 2 π 2 4 w 1 t -k 2 π 2 4 w dt cosh 2 √ t   = +∞ k 2 π 2 4 ε ′ q 3 (t) cosh 2 √ t dt (3.50) with q 3 (t) := 2 kπ t 0 ε ′ 1 w 1 -y(1 + w) 1 √ t 0 -w ✶ {w< 1-y y } dw and t 0 = 4t k 2 π . 1) Observe that w → - 1 √ 1 -y ln √ 1 -y + 1 -y(1 + w) √ 1 -y -1 -y(1 + w) is a primitive of 1/w 1 -y(1 + w), therefore 1-y y ε ′ dw w 1 -y(1 + w) = 1 √ 1 -y ln √ 1 -y + 1 -y(1 + ε ′ ) √ 1 -y -1 -y(1 + ε ′ ) = 1 √ 1 -y ρ 1 (ε ′ )
and

q 1 = 1 √ 1 -y ρ 1 (ε ′ ) +∞ 0 kds √ s cosh 2 k s + π 2 4 ε ′ (3.51)
after the change of variable v = k 2 s.

2) Now we simplify q 3 (t). Let u = (t 0w)/(1y(1 + w)). From this change of variable and we get:

• if t 0 > (1 -y)/y, q 3 (t) = 4 kπ +∞ u 0 du u 2 (1 -y) -t 0 = 1 t(1 -y) ln   u 0 + t 0 1-y u 0 - t 0 1-y   , • if t 0 < (1 -y)/y, q 3 (t) = 4 kπ 0 u 0 du u 2 (1 -y) -t 0 = 1 t(1 -y) ln   u 0 + t 0 1-y t 0 1-y -u 0   where u 0 = (t 0 -ε ′ )/(1 -y(1 + ε ′ )).
We make the change of variable t = k 2 (s + π 2 ε ′ /4) in (3.50), then t 0 /(1y) = u 2 (s, ε ′ ), u 0 = u 1 (s, ε ′ ) and

q 3 (t) = 1 k 1 s + π 2 ε ′ 4 ρ 2 (s, ε ′ ).
Finally, (3.47) is a consequence of (3.51), (3.50) and the above identity.

Using identities (3.44) and (3.47), the definition (2.11) of h in and interverting (legally) the integral and the sum gives the following corollary because the cutoff ε ′ permits to intervert the integral and the sum. Finally, we can write p g * (y) as a limit of an integral of a function which is expressed as a series.

Corollary 3.11. We have

p g * (y) = 1 2π y(1 -y) lim ε ′ →0 +∞ 0   ρ 1 (ε ′ ) √ s - ρ 2 (s, ε ′ ) s + π 2 4 ε ′   h s + π 2 4 ε ′ ds (3.52)
where h, ρ 1 and ρ 2 have respectively been defined in (2.11) and (3.48).

Here we compute the limit of the term ρ 1 (ε ′ )/ √ sρ 2 (s, ε ′ )/ s + π 2 ε ′ /4 appearing in the previous integral and we verify that the dominated convergence theorem may be applied.

Lemma 3.12. For any s > 0, lim

ε ′ →0   ρ 1 (ε ′ ) √ s - ρ 2 (s, ε ′ ) s + π 2 4 ε ′   h s + π 2 4 ε ′ = ln 1 - π 2 (1 -y) 4ys h( √ s) √ s . (3.53)
Proof of Lemma 3.12 We use the following decomposition

ρ 1 (ε ′ ) √ s - ρ 2 (s, ε ′ ) s + π 2 4 ε ′ = ρ 3 (ε ′ ) √ s - ρ 4 (s, ε ′ ) s + π 2 4 ε ′ + ρ 5 (s, ε ′ ) s + π 2 4 ε ′ +ln ε ′   1 s + π 2 4 ε ′ - 1 √ s   (3.54)
where

ρ 3 (ε ′ ) = 2 ln 1 -y + 1 -y(1 + ε ′ ) -ln(y), (3.55) ρ 4 (s, ε ′ ) = 2 ln (u 1 (s, ε ′ ) + u 2 (s, ε ′ )) + ln π 2 (1 -y) + ln (1 -y(1 + ε ′ )) -ln(4y), (3.56) ρ 5 (s, ε ′ ) = ln s - π 2 4 1 -y y + π 2 4 ε ′ . (3.57)
It is easy to deduce from (3.49) and (2.11):

lim ε ′ →0 ρ 3 (ε ′ ) = 2 ln 2 + ln(1 -y) -ln(y), lim ε ′ →0 ρ 4 (s, ε ′ )/ s + π 2 4 ε ′ = ln(4s(1 -y)/y)/ √ s, lim ε ′ →0 ρ 5 (s, ε ′ ) = ln s -π 2 (1 -y)/4y , lim ε ′ →0 h s + π 2 4 ε ′ = h( √ s).
Then (3.53) follows from (3.54).

Proof of (2.12)

We claim that we can intervert in (3.52) the limit (ε ′ → 0) and the integral, using the dominated convergence theorem. Then (2.12) is a direct consequence of Lemma 3.12. Indeed, there exists a generic constant C > 0 (that may change from one line to another) such that we have the following inequalities:

|ρ 3 (ε ′ )| √ s C √ s , ∀s > 0, ∀ε ′ ∈ (0, 1)., |ρ 4 (s, ε ′ )| s + π 2 4 ε ′ C √ s (1 + ln(1 + s) + | ln s|) , ∀s > 0, ∀ε ′ ∈ (0, 1], |ρ 5 (s, ε ′ )| s + π 2 4 ε ′ C √ s 1 + ln(1 + s) + ln s - π 2 (1 -y) 4y , s ∈ I c ε ′
where

I c ε ′ := 0, π 2 (1 -y) 4y -π 2 ε ′ ∪ π 2 (1 -y) 4y , +∞ .
It remains to investigate the limit of the integral when s ∈ I ε ′ . For y > 0 fixed and ε ′ small enough,

1 s + π 2 4 ε ′ C, h s + π 2 4 ε ′ C, s ∈ I ε ′
The change of variable u = sπ 2 (1y)/4y leads to:

+∞ 0 ρ 5 (s, ε ′ ) s + π 2 4 ε ′ h s + π 2 4 ε ′ C π 2 4 ε ′ -3π 2 4 ε ′ | ln u|du. (3.58)
Consequently the left hand side in (3.58) goes to 0 as ε ′ → 0. The proof of (2.12) is now complete.

Proof of (2.13) in Theorem 2.1

Our purpose is to get rid off the series that appears in the definition of h.

In order to keep the guiding principal, some technical points are proved in Section 3.4. Our proof has five main steps, numbered from 1 to 5.

1)

Starting with (2.12), our goal is to simplify

ρ(y) := +∞ 0 ln 1 - π 2 (1 -y) 4ys h( √ s) √ s ds.
In order to switch the sum and the integral, we need to determine the behavior of h in the vicinity of 0 and +∞.

Lemma 3.13. One has (i) for any x > 0, |h(x)| 4e -2x /(1e -2x ) 2 ;

(ii) for any x ∈ (0, 1), |h(x)| C/ √ x where C is a positive constant.

The proof of Lemma 3.13 is postponed in Section 3.4. The simplification of ρ is based on the following identity:

ln 1 - π 2 (1 -y) 4ys = ln π 2 (1 -y) 4y -ln s + ln 4ys π 2 (1 -y) -1 . (3.59)
This gives rise to a decomposition of ρ(y) as the sum of three terms.

Lemma 3.14. One has

ρ(y) = a 1 ln π 2 (1 -y) 4y + a 2 + τ (y) (3.60)
where

a 1 := +∞ 0 h( √ s) √ s ds, a 2 := - +∞ 0 ln(s) h( √ s) √ s ds, (3.61)
and τ (y) :

= +∞ 0 ln 4ys π 2 (1 -y) -1 h( √ s) √ s ds. (3.62)
We calculate τ (Lemma 3.19), a 1 (item 3)) and a 2 (Lemma 3.21).

2)

We begin with τ . We can eliminate the sum in h( √ s) but only on [ε, +∞[.

Let us introduce

ϕ(a, n) := n i=1 a 2 k 2 -1 (-1) k+1
, a > 0, n 1.

(3.63) Lemma 3.15. For any y > 0, τ (y) = lim ε → 0 τ 1 (ε, y) where:

τ 1 (ε, y) := +∞ ε ln ϕ 4yt π 2 (1 -y) , t ε dt √ t cosh 2 ( √ t) . ( 3

.64)

Proof of Lemma 3.15 a) First, using (ii) of Lemma 3.13, we get that the function s → ln 4ys

π 2 (1-y) -1 h( √ s) √ s is integrable on [0, ∞], consequently τ (y) = lim ε → 0 τ 1 (ε, y) with τ 1 (ε, y) = +∞ ε ln 4ys π 2 (1 -y) -1 h( √ s) √ s ds. (3.65) b)
Second, it is possible to permute the integral and the sum in τ 1 (ε, y) (Equation (3.65)). In that view, we will use the following result: let (f k ) k 1 be a sequence of functions defined on [0, +∞[ and such that

k 1 +∞ 0 |f k (s)|ds < +∞. (3.66) Then +∞ 0 k 1 f k (s) ds = +∞ 0 k 1 f k t k 2 1 k 2 dt. (3.67)
Now let us define

f k (s) := (-1) k+1 ln 4ys π 2 (1 -y) -1 k √ s cosh 2 (k √ s)
✶ {s ε} .

Since cosh x e x /2 and √ s √ ε, we get

k 1 |f k (s)| ln 4ys π 2 (1 -y) -1 4 √ ε k 1 ke -2k √ s 4 √ ε ln 4ys π 2 (1 -y) -1 e -2 √ s (1 -e -2 √ ε ) 2 using k 1 kρ k = ρ d dρ k 1 ρ k = ρ (1 -ρ) 2 .

Noticing that

∞ 0 e -2 √ s ln 4ys π 2 (1-y) -1 ds < ∞, (3.66) holds, we may apply (3.67):

τ 1 (ε, y) = +∞ 0 k 1 (-1) k+1 ln 4yt π 2 (1 -y)k 2 -1 ✶ {k 2 t ε } 1 √ t cosh 2 ( √ t) dt. Since k 1, k 2 ε t implies t ε and k 2 ε t ⇔ k √ t/ε ⇔ k ⌊ √ t/ε⌋. We introduce a := 4yt/(π 2 (1 -y)), n := ⌊ √ t/ε⌋, then n k=1 (-1) k+1 ln a 2 k 2 -1 = ln |ϕ(a, n)|
from which we deduce the Lemma 3.15.

To determine the limit of τ 1 (y, ε) in (3.65), we calculate the limit of ϕ(a, m) when m → ∞ (cf. Lemma 3.17). We prove in Lemma 3.18 that the dominated convergence theorem applies. We begin with modifying ϕ(a, m) so that its limit as m → ∞ can be calculated.

Lemma 3.16. One has

ϕ(a, 2n) = Γ(n + 1 -(a + 1)/2)Γ(n + 1 + (a -1)/2) Γ(n + 1 -a/2)Γ(n + 1 + a/2) Γ(n + 1) Γ(n + 1/2) 2 α(a)
where α(a)

:= π Γ(1 -a/2)Γ(1 + a/2) Γ(1 -(a + 1)/2)Γ(1 + (a -1)/2)
.

Proof of Lemma 3.16 By definition, we get

ϕ(a, 2n) = n k=1 k -a+1 2 k -a 2 n k=1 k + a-1 2 k + a 2 n k=1 k k -1 2 2 .
Recalling that Γ(x + 1) = xΓ(x) for all x / ∈ N, we easily deduce

n k=1 (k + b) = Γ(n + 1 + b) Γ(1 + b) and ϕ(a, 2n) = Γ(n + 1 -(1 + a)/2)Γ(n + 1 + (a -1)/2) Γ(n + 1 -a/2)Γ(n + 1 + a/2) Γ(n + 1) Γ(n + 1/2) 2 α(a) with α(a) = (Γ(1/2)) 2 Γ(1 -a 2 )Γ(1 + a 2 )/ Γ(1 -1+a 2 )Γ(1 + a-1 2 ) . It remains to notice that Γ(1/2) =
√ π to conclude the proof.

We are now able to compute the limit of ϕ(a, m) as m goes to infinity.

Lemma 3.17. For any a,

lim m→+∞ |ϕ(a, m)| = |α(a)| = πa 2 cot πa 2 .
Proof of Lemma 3.17 (i) Assume that m = 2n. Recall that Γ(1 + x) ∼ √ 2πx x+1/2 e -x as x → ∞ and notice that

n -(a + 1)/2 n -a/2 = 1- 1 2n +o(1/n) and (n-a/2) ln n -(a + 1)/2 n -a/2 → -1/2. Then Γ(n + 1 -(a + 1)/2) Γ(n + 1 -a/2) ∼ 1/ √ n, n → ∞.
Changing the variable a in -a leads to

Γ(n + 1 + (a -1)/2) Γ(n + 1 + a/2) ∼ 1/ √ n, n → ∞.
Similarly, we get Γ(n + 1)

Γ(n + 1/2) ∼ √ n, n → ∞.
Consequently, lim n→+∞ ϕ(a, 2n) = α(a) (ii) Assume now that m = 2n + 1. It suffices to notice that ϕ(a, 2n + 1) = ϕ(a, 2n)

a 2
(2n+1) 2 -1 and to apply the previous result to get that the limit of ϕ(a, 2n + 1) is -α(a) when n → +∞.

(iii) Finally, it remains to simplify α(a). Using the identities Γ(1+x) = xΓ(x) with x = a/2 and Γ(z)Γ(1z) = π/ sin(πz), z / ∈ -N for z = a/2 and z = 1/2a/2 yields the required result.

Since

t/ε → +∞ as ε → 0, then Lemma 3.17 implies:

lim ε→0 ln ϕ 4ty π 2 (1 -y) , t ε = ln α 4ty π 2 (1 -y) = ln ty 1 -y cot ty 1 -y .
It remains to apply the dominated convergence theorem to get the limit of τ 1 (y, ε). Its legal use is justified by the following lemma.

Lemma 3.18. (i) First,

ln ϕ 4yt π 2 (1 -y) , t ε ϕ 1 (t) with ϕ 1 (t) := k 1 ln 1 - t a 3 k 2 and a 3 := π 2 (1 -y) 4y .
(ii) Second,

A := ∞ 0 ϕ 1 (t) dt √ t cosh 2 ( √ t) < ∞. (3.68) Proof of Lemma 3.18 (i) By definition, we have ln |ϕ(a, n)| = n k=1 (-1) k+1 ln |1 -a 2 /k 2 | and thus ln |ϕ(a, n)| n k=1 ln 1 - a 2 k 2 .
(ii) We decompose A defined by (3.68) as

A = A 1 + A 2 with A 1 = a 3 /2 0 ϕ 1 (t) dt √ t cosh 2 ( √ t) and A 2 = ∞ a 3 /2 ϕ 1 (t) dt √ t cosh 2 ( √ t) .
We prove that A 1 and A 2 are finite.

• Since there exists c > 0 such that | ln(1

-x)| cx for 0 < x < 1/2, condition t a 3 /2 implies that ln 1 - t a 3 k 2 ct a 3 k 2 .
from which we deduce that

A 1 c a 3 k 1 1 k 2 a 3 /2 0 √ tdt cosh 2 ( √ t) < ∞ .
• Since ϕ 1 (t) is a series with positive terms, we have:

A 2 = k 1 ∞ a 3 /2 ln 1 - t a 3 k 2 dt √ t cosh 2 ( √ t) .
and the change of variable t = k 2 s leads to:

A 2 = k 1 k ∞ a 3 /2k 2 ln 1 - s a 3 ds √ s cosh 2 (k √ s) .
Now since cosh x e x /2, we get

A 2 4 ∞ 0 ln 1 - s a 3 1 √ s k 1 ke -2k √ s ds 4 ∞ 0 ln 1 - s a 3 e -2 √ s (1 -e -2 √ s ) 2 ds √ s
Finally we conclude that the integral converges at infinity and 0.

We have established the following lemma.

Lemma 3.19. The function τ can be simplified as:

τ (y) = 2 +∞ 0 ln s y 1 -y cot s y 1 -y ds cosh 2 (s)
.

(3.69)

3) Let us prove that a 1 = 1. We use the following probabilistic result.

Lemma 3.20. Let (ζ(α)) 0<α<1 be a sequence of geometrical random variables with parameter α. Then η(α

) := (1 -α) ζ(α) converges in distribution to U ([0, 1]) as α → 0.
Admit for a while Lemma 3.20 whose proof is postponed at the end of item 3).

First, by Lemma 3.13,

a 1 = lim ε→0 a 1 (ε) where a 1 (ε) := +∞ ε h( √ s) √ s ds.
By permuting the sum and the integral, letting k √ s = t and introducing ν = e -2k √ ε , we obtain:

a 1 (ε) = 2 +∞ k=1 (-1) k+1 +∞ k √ ε dt cosh 2 t = 2 +∞ k=1 (-1) k+1 1 -tanh(k √ ε) = 4 +∞ k=1 (-1) k+1 e -2k √ ε 1 + e -2k √ ε = 4 +∞ k=1 (-1) k+1 ν k 1 + ν k = 4 +∞ k=1 ν 2k-1 (1 -ν) (1 + ν 2k-1 )(1 + ν 2k ) ,
where the last equality comes from separating the odd and even numbers. Consequently,

a 1 (ε) = 4ν 1 + ν +∞ k=1 1 (1 + 1 ν ν 2k )(1 + ν 2k ) P(ζ(1 -ν 2 ) = k) = 4ν 1 + ν E ϕ 1 ν , τ (1 -ν 2 ) where ϕ (u, v) = 1 (1 + uv)(1 + v) , u, v 0.
Since ϕ is continuous, bounded and ν → 1 as ε → 0, it remains to apply Lemma 3.20 to get

a 1 = lim ε→0 a 1 (ε) = 2 1 0 du (1 + u) 2 = 1.
Proof of Lemma 3.20 Let 0 < u < 1. Then We conclude to (2.13) and the proof Theorem of 2.1 .

P(η(α) < u) = P ζ(α) > ln u ln(1 -α) = (1 -α) ⌊ln u/ ln(1-α)⌋ → u, as α → 0.

Technical results

In this subsection, we prove the intermediate results used in Section 3, namely Lemma 3.13, and identities (3.72) and (3.73).

Proof of Lemma 3.13 (i)

The first inequality is direct as soon as one notices that cosh x e x /2.

(ii) Let h 0 and k be the functions defined by h 0 (x) := xh(x) for x > 0 and h 1 (x) := x/ cosh 2 (x) for x 0. The Taylor expansion of h 1 with integral rest can be written in the following form:

h 1 (b) -h 1 (a) = (b -a)h ′ 1 (b) + R(a, b) where R(a, b) := (b -a) 1 0 (h ′ 1 (a + r(b -a)) -h ′ 1 (b))dr. Then, h 0 (x) = k 1 (-1) k+1 h 1 (kx) = -(h 01 (x) + h 02 (x) + h 03 (x))
where

h 01 (x) = x 1 n x * h ′ 1 (2nx) h 02 (x) = 1 n x * R((2n -1)x, 2nx) h 03 (x) = n>x * (h 1 (2nx) -h 1 ((2n -1)x))
and x * = x -3/2 . The derivative of h 1 being bounded, we deduce that |h 02 (x)| c √ x. Since h 1 (x) ce -x , we can easily prove that |h 03 (x)| ce -2/ √ x /x. As for h 01 (x), we decompose this sum with the related integral:

h 01 (x) = x 0 h ′ 1 (2y)dy + h01 (x) where h01 (x) = x 1 n nx nx (n-1)x (h ′ 1 (2nx) -h ′ 1 (2y))dy , n x = ⌊x * ⌋ and x = n x x. From the previous estimates, we have | h01 (x)| c √ x and x 0 h ′ 1 (2y)dy = 1 2 h 1 (x) ce -2/ √ x . 29
2k+1) √ ε 2k √ ε f (t)dt. (
• Obviously, we first have

a 21 (ε) + a ′ 21 (ε) = 4 +∞ √ ε f (t)dt a 21 (ε) -a ′ 21 (ε) = 4 +∞ k=1 2k √ ε (2k-1) √ ε (f (t) -f (t + √ ε))dt.
• Second, we prove that lim ε→0 (a 21 (ε)a ′ 21 (ε)) = 0. Indeed, we notice that for small values of ε, we have f

(t + √ ε) -f (t) 0, t ∈ [0, 1] since for all t ∈ [0, 1], f ′ (t) > 0. Then |a 21 (ε) -a ′ 21 (ε)| 4 1 0 (f (t + √ ε) -f (t))dt + 4 +∞ 1 f (t + √ ε) -f (t) dt.
On one hand,

1 0 (f (t + √ ε) -f (t))dt = 1+ √ ε 1 f (t)dt - √ ε 0 f (t))dt -→ ε→0 0 since f is integrable on R + .
On the other hand, for any t 1, |f ′ (t)| Ce -t that leads to

+∞ 1 f (t + √ ε) -f (t) dt C √ ε +∞ 1 e -t dt -→ ε→0 0.
• Finally, we derive (3.72) from the previous results and the following equation We begin with a new decomposition of a 22 (ε).

a 21 (ε) = 1 2 (a 21 (ε) + a ′ 21 (ε)) + 1 2 (a 21 (ε) -a ′ 21 (ε)). 3
Lemma 3.22.

a 22 (ε) = 4 +∞ √ ε g 1 ε (t) + g 2 ε (t) + O 1 n (θ(t + √ ε) + θ(t)) dt (3.75)
where Using Stirling formula

g 1 ε (t) = n ln n θ(t + √ ε) -θ(t) + 1 2 ln nθ(t + √ ε), g 2 ε (t) = (ln 2 -1)n θ(t + √ ε) -θ(t) ln √ 2πθ(t + √ ε) - 1 2 ln 2θ(t).
k ! = √ 2πk k+1/2 exp -k + η(k) 12k , k 1
where 1 -1/(12k + 1) η(k) < 1, we easily deduce the asymptotic behavior of S e m and S o m as m → +∞:

S e k = k ln k + (ln 2 -1)k + 1 2 ln k + ln √ 2π + O 1 k , (3.76)
S o k = k ln k + (ln 2 -1)k + 1 2 ln 2 + O 1 k (3.
+∞ 2k √ ε θ(t)dt = n k=1 ln(2k) +∞ (2k-1) √ ε θ(t + √ ε)dt = +∞ √ ε n k=1 ln(2k)✶ {(2k-1) √ ε t} θ(t + √ ε)dt = +∞ √ ε S e n θ(t + √ ε)dt and similarly, n k=1 ln(2k -1) +∞ (2k-1) √ ε θ(t)dt = +∞ √ ε S o n θ(t)dt.
Then (3.75) is a direct consequence of (3.71), (3.76) and (3.77).

By Lemma 3.22, a 22 (ε) is the sum of three terms. We determine the limit of each term separately.

1) It is clear that We calculate θ 1 and the limit of θ 2 (ε) as ε → 0.

lim ε→0 +∞ √ ε O 1 n (θ(t + √ ε) + θ(t))dt = 0. (3.78) 2) Note that n|θ(t + √ ε) -θ(t)| Ce -t , t 0, ε > 0. ( 3
(i) By an integration by parts, it comes (ii) We determine the limit of θ 2 (ε), ε → 0. Noticing that

+∞ √ ε t θ(t + √ ε) -θ(t) √ ε dt = 1 √ ε +∞ 2 √ ε (t - √ ε)θ(t)dt - +∞ √ ε tθ(t)dt = - +∞ √ ε θ(t)dt - 1 √ ε 2 √ ε √ ε (t - √ ε)θ(t)dt.
Consequently,

θ 2 (ε) = 1 2 √ ε ln(2 √ ε) 2 √ ε √ ε (t - √ ε)θ(t)dt

Figure 1 :

 1 Figure 1: The r.v.'s g(1), U (1), U * (1), f * (1) and g *

. 11 )

 11 Theorem 2.1. [Probability distribution function of g * ]

1 g

 1 (s); 0 s 1) represents the standard Brownian bridge and 1 g(1) B(sg(1)); 0 s 1 is independent of g(1). (2.15) If we replace U by |b| in the scheme (1.7)-(1.10) (resp. (1.7)-(1.8)), we get g b (resp. b * (1)) which have been plotted in Figure 2

Figure 2 :

 2 Figure 2: The r.v.'s g b and b * (1)

E [ FCorollary 3 . 2 .

 [32 (b(u); 0 u 1)] = √ 2πE[F (br * (u); 0 u 1)M * ] structure of the trajectory of br * and (3.25) to give a first expression of the distribution of g b in terms of Brownian random variables. Let f be a bounded Borel function, then

3 .27) where B(σ 1 ) := inf 0 u σ 1 B

 311 (u) and B(σ 1 ) := inf 0 u σ1 B(u). Proof of Corollary 3.2 First, notice that b * (1) = sup 0 u 1 |b(u)| = b(1) ∨ b(1), where b(1) := sup 0 u 1 b(u) and b(1) :=inf 0 u 1 b(u). Moreover, b * (1) = b(1) ⇔ b(1) > b(1) and on that set, b * (1) = b(1). Since (-b(u); 0

Figure 3 :

 3 Figure 3: Notation br * (1) and ρ

Figure 4 :

 4 Figure 4: Scaling

  .30) Using (3.28), Corollary 3.2 and Lemma 3.3, we deduce: Corollary 3.4. Let f be a bounded Borel function, then

5 )

 5 Finally, we are able to compute ρ(y). Plugging (3.69) into (3.60), using a 1 = 1 and (3.70), we get ρ(y) Setting x = s y/(1y), we have:

  77)where O (1/k) stands for a sequence bounded by 1/k.

implies θ 1 =

 1 -1/2.

and lim ε→0 θ 2

 2 (ε) = 0. Moreover using (3.89) and θ 1 = -1/2, the limits (3.78), (3.80) and (3.92) into (3.75) implies (3.73) which completes the proof.

  .34) Proof of Lemma 3.7 Let f : R → R be a bounded Borel function. Using Proposition 2.2, Corollary 2.4 and (2.18), we obtain:

4 )

 4 We calculate a 2 . We admit two intermediate results (whose proofs are postponed in Section 3.4) so that the reader can have quickly the explicit value of a 2 .

	Proof of Lemma 3.21 We proceed as for the calculation of a 1 and we
	straightforwardly get a 2 = -lim ε→0	ε	+∞	h( √ s √ s)	ln(s)ds = -lim ε→0	(a 21 (ε) + a 22 (ε))
	where a 21 (ε) := 4	+∞ k=1	2k (2k-1) √ ε	√ ε	ln t cosh 2 (t)	ds,
	a 22 (ε) :=	+∞ k=1	ln(2k)	+∞ 2k √ ε	dt cosh 2 (t)	-ln(2k -1)	+∞ (2k-1)	√ ε	dt cosh 2 (t)	.
	Lemma 3.21. We have
				a 2 = -2	0	+∞	ln t cosh 2 (t)	dt -ln	π 2 4	.	(3.70)

  √ ε as ε goes to 0 and thus n (θ(t + √ ε)θ(t)) ∼ t 2 θ ′ (t), ε → 0. We apply the Lebesgue's dominated convergence theorem and we get We are interested in the limit of the integral involving g 1 ε (t). In that view, we decompose g 1 ε (t) in Lemma 3.23 using a new class of functions that we define as follows. The sequence of functions (R ε (t), t 0) ε∈(0,1/2] is said to belong to the class R if|R ε (t)| dt < +∞.. (3.82) Clearly, if (R ε (t), t 0) ε∈(0,1/2] belongs to the class R then lim ε→0Proof of Lemma 3.23 The function g 1 ε (defined in Lemma 3.22) is the sum of tow terms. We study each of them separately. a) We prove that

	Finally, Using the definition (3.74) of n, we have n -t lim ε→0 2 √ ε +∞ √ ε g 2 ε (t)dt = 1 2 (ln π + 1 -ln 2) . 1 2 . Then λ(n) -λ t 2 √ ε 1 2 | sup u∈J |λ ′ (u)| 1 2 1 + ln 1 + t √ ε	(3.80) (3.86)
	where J is the interval with end points n and t/2 θ(t + √ ε) -θ(t) C √ εe -t . √ ε. It is clear that Then (3.81) and (3.82) follow from (3.85), (3.86) and (3.87). 3) (i) lim ε→0 R ε (t) = 0, t 0, (ii) +∞ 0 ln(t/2 √ ε)) = 0 using (3.74). Using moreover (3.87), we deduce that sup 0<ε<1/2 +∞ ln(n)θ(t + √ ε) = ln t 2 θ(t) + R ε (t). √ ε 0 0. c) Then (3.83) follows from (3.84), (3.88), the following identity	(3.87) (3.81) (3.88) R ε (t)dt =
	Lemma 3.23. One has g 1 ε (t) = 1 2 (t ln(t)θ ′ (t) + ln(t)θ(t))-t ln t 2 √ ε θ(t + √ ε) -θ(t) = 1 2 ln(2 √ and easy calculations.	ε) t t ln t 2	θ(t + θ ′ (t) + R ε (t) √ ε) -θ(t) √ ε	+ θ(t) +R ε (t).
	According to Lemma 3.23,				(3.83)
	+∞
	√ ε	g 1 ε (t)dt = θ 1 + θ 2 (ε) + o(ε)	(3.89)
	.79) ε) -θ(t) + R ε (t) (3.84) where n has been defined in (3.74) and (R ε (t), t 0) ε∈(0,1/2] ∈ R. It suffices Moreover, n ∼ t 2 n ln n θ(t + √ ε) -θ(t) = t 2 √ ε ln t 2 √ ε θ(t + where √ to show that θ 1 := +∞ 1 (t ln(t)θ ′ (t) + ln(t)θ(t)) dt, (3.90) 2 0 θ 2 (ε) := -1 2 ln(2 √ ε) +∞ √ ε t θ(t + √ ε) -θ(t) √ ε + θ(t) dt. (3.91)
	lim ε→0 R ε (t) = n ln n θ(t + +∞ √ ε n θ(t + √ ε) -θ(t) dt = = √ ε) -θ(t) -1 2 1 2 2 = λ(n) -λ(t/2 √ ε) θ(t + √ ε) -θ(t) +∞ 0 tθ ′ (t)dt [tθ(t)] +∞ 0 -t √ ε ln t 2 √ ε belongs to R where λ(x) = x ln x.	0	+∞ θ(t + θ(t)dt = -√ ε) -θ(t) 1 2 (3.85) .

b) Since | ln aln b| |a -b|/ min{a, b}, a, b > 0, we have lim ε→0 (ln n -