Javier Quílez-Bermejo

Sara Pérez-Rodríguez

Rafael Canevesi

Daniel Torres

Emilia Morallón

Diego Cazorla-Amorós

Alain Celzard

Vanessa Fierro
email: vanessa.fierro@univ-lorraine.fr

Easy enrichment of graphitic nitrogen to prepare highly catalytic carbons for oxygen reduction reaction

Keywords: oxygen reduction reaction, electrocatalysts, porous carbons, nitrogen, functionalization, graphitic enrichment

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The oxygen reduction reaction (ORR) has a huge impact on the development of many electrochemical devices, such as fuel cells or metal-air batteries [START_REF] Wang | Compositing doped-carbon with metals, non-metals, metal oxides, metal nitrides and other materials to form bifunctional electrocatalysts to enhance metal-air battery oxygen reduction and evolution reactions[END_REF][START_REF] Li | Relating Catalysis between Fuel Cell and Metal-Air Batteries[END_REF][START_REF] Morozan | Low-platinum and platinum-free catalysts for the oxygen reduction reaction at fuel cell cathodes[END_REF][START_REF] Quílez-Bermejo | Metal-free heteroatom-doped carbon-based catalysts for ORR. A critical assessment about the role of heteroatoms[END_REF]. This reaction is considered a bottleneck in these technologies due to its sluggish kinetics and large overpotentials [START_REF] Lang | Recent Advances and Prospects of Metal-Based Catalysts for Oxygen Reduction Reaction[END_REF][START_REF] Gasteiger | Activity benchmarks and requirements for Pt, Pt-alloy, and non-Pt oxygen reduction catalysts for PEMFCs[END_REF]. The commercial electrodes that are commonly used to catalyze ORR are extremely expensive to make this forefront equipment competitive in comparison with current fossil fuel-based engines [START_REF] Sealy | The problem with platinum[END_REF][START_REF] Dai | Metal-Free Catalysts for Oxygen Reduction Reaction[END_REF]. The most common commercial ORR electrocatalysts are based on platinum nanoparticles supported on carbon materials [START_REF] Morozan | Low-platinum and platinum-free catalysts for the oxygen reduction reaction at fuel cell cathodes[END_REF][START_REF] Sealy | The problem with platinum[END_REF][START_REF] Dai | Metal-Free Catalysts for Oxygen Reduction Reaction[END_REF]. Platinum is a rare material in nature, which drives the cost of the cathode to unsustainable levels. Moreover, these metal nanoparticles lixiviate and agglomerate under working conditions, reducing the catalytic activity over time [START_REF] Ohyagi | Effects of operating conditions on durability of polymer electrolyte membrane fuel cell Pt cathode catalyst layer[END_REF][START_REF] Winter | What are batteries, fuel cells, and supercapacitors?[END_REF].

Several materials have been presented as attractive alternatives to platinum-based catalysts [START_REF] Lang | Recent Advances and Prospects of Metal-Based Catalysts for Oxygen Reduction Reaction[END_REF][START_REF] Zhou | A review of graphene-based nanostructural materials for both catalyst supports and metal-free catalysts in PEM fuel cell oxygen reduction reactions[END_REF][START_REF] Xue | Transition metal oxide-based oxygen reduction reaction electrocatalysts for energy conversion systems with aqueous electrolytes[END_REF]. It is worth noting the significant findings in recent years on carbon materials, which have emerged as a promising option to develop low-cost ORR electrodes, given that carbon and carbon precursors are abundant in nature and that carbon-based electrodes are affordable for large-scale implementation [START_REF] Quílez-Bermejo | Metal-free heteroatom-doped carbon-based catalysts for ORR. A critical assessment about the role of heteroatoms[END_REF][START_REF] Wu | A Discussion on the Activity Origin in Metal-Free Nitrogen-Doped Carbons for Oxygen Reduction Reaction and their Mechanisms[END_REF][START_REF] Seredych | Electrochemical Reduction of Oxygen on Hydrophobic Ultramicroporous PolyHIPE Carbon[END_REF].

In general, it is known that the catalytic activity of carbon materials is related to the amount and chemical nature of the active sites [START_REF] Ji | The progress of metal-free catalysts for the oxygen reduction reaction based on theoretical simulations[END_REF]. Undoped carbon materials have been shown to be far from the commercial electrocatalysts due to the homogeneous charge distribution along the carbon layers that does not facilitate the chemisorption of dioxygen molecules [START_REF] Zheng | Nanostructured metal-free electrochemical catalysts for highly efficient oxygen reduction[END_REF].

Nevertheless, surface area and well-controlled pore size distribution of carbon materials play an important role in the ORR catalysis. Indeed, a large surface area with well-developed microporosity has been shown to be favorable for achieving exceptional ORR activities [START_REF] Yang | Carbon-Based Metal-Free ORR Electrocatalysts for Fuel Cells : Past, Present, and Future[END_REF][START_REF] Gabe | Modeling of oxygen reduction reaction in porous carbon materials in alkaline medium. Effect of microporosity[END_REF]. Despite this, the introduction of defects or heteroatoms, in particular nitrogen (N), is crucial to achieve excellent catalytic properties in carbon-based catalysts [START_REF] Quílez-Bermejo | Metal-free heteroatom-doped carbon-based catalysts for ORR. A critical assessment about the role of heteroatoms[END_REF]. In this sense, some Ndoped carbon materials have been shown to catalyze the ORR with almost platinum-like performance [START_REF] Gong | Nitrogen-doped carbon nanotube arrays with high electrocatalytic activity for oxygen reduction[END_REF][START_REF] Quílez-Bermejo | Oxygen reduction catalysis of Ndoped carbons prepared via heat treatment of polyaniline at over 1100 o C[END_REF], although the origin of this high catalytic activity is still unclear.

Important findings have been made in recent years, leading to a general understanding of the active sites of these carbon-based catalysts; the introduction of nitrogen atoms into the carbon layers results in an electron-withdrawal effect towards the adjacent carbon atoms, creating a positive charge density that facilitates chemisorption of dioxygen molecules and their reduction [START_REF] Wiggins-Camacho | Mechanistic Discussion of the Oxygen Reduction Reaction at Nitrogen-Doped Carbon Nanotubes[END_REF][START_REF] Quílez-Bermejo | On the origin of the effect of pH in ORR for non-doped and edge-type quaternary Ndoped metal-free carbon-based catalysts[END_REF][START_REF] Maldonado | Influence of nitrogen doping on oxygen reduction electrocatalysis at carbon nanofiber electrodes[END_REF]. Although knowledge about the origin of catalytic activity is increasing, still some controversy exists about which N functional groups create the highest activities towards the ORR. Most scientists agree that pyridines and graphitic nitrogen seem to be the most promising species [START_REF] Wu | A Discussion on the Activity Origin in Metal-Free Nitrogen-Doped Carbons for Oxygen Reduction Reaction and their Mechanisms[END_REF][START_REF] Singh | Active Sites and Mechanism of Oxygen Reduction Reaction Electrocatalysis on Nitrogen-Doped Carbon Materials[END_REF][START_REF] Liu | Recent Advances in Heteroatom-Doped Metal-Free Electrocatalysts for Highly Efficient Oxygen Reduction Reaction[END_REF]. Nevertheless, due to the coexistence of different nitrogen species in most carbon materials, the individual evaluation of their catalytic activity is difficult. Recent results obtained with carbon materials selectively doped with graphitic nitrogen show high catalytic activity, close to that for platinum [START_REF] Quílez-Bermejo | Towards understanding the active sites for the ORR in N-doped carbon materials through fine-tuning of nitrogen functionalities: an experimental and computational approach[END_REF][START_REF] Quílez-Bermejo | Polyaniline-derived N-doped Ordered Mesoporous Carbon Thin Films: Efficient Catalysts towards Oxygen Reduction Reaction[END_REF]. However, N-doped carbon materials that have been obtained with purely graphitic N require complex methodologies and the use of N-containing carbon precursors, making it impossible to extend the synthetic route to other carbon materials. Therefore, post-functionalization methodologies for purely graphitic N enrichment are mandatory in order to selectively introduce graphitic N species into multiple carbon materials while retaining most of the properties of the pristine carbon materials. Graphitic N can further improve the performance of carbon materials not only for ORR electrocatalysts, but also for multiple applications.

In this study, highly efficient metal-free carbon-based catalysts have been synthesized from selective doping with graphitic nitrogen of a well-known commercial activated carbon and using a straightforward post-functionalization methodology. This high-surface area commercial carbon was doped with urea at different carbon to urea mass ratios, and heattreated at several temperatures to tune its surface chemistry. Some functionalization treatments led to very efficient catalysts for ORR, approaching platinum-based electrocatalysts. The extensive characterization and electrocatalytic results provided important findings that corroborate the role of graphitic nitrogen in the catalytic activity towards ORR of N-doped carbon materials.

Materials and methods

Preparation of nitrogen-doped activated carbons

A commercial activated carbon, MSC30 (Kansai Coke and Chemicals Co., Ltd. Japan), was functionalized using urea as a nitrogen precursor by a two-step procedure. First, MSC30 and urea were physically mixed in mass ratios of 1:1 and 1:2 and subjected to heat treatment in a quartz boat up to 350 ºC for 3 h with a heating rate of 5 ºC min -1 in synthetic air (50 mL min - 1 ). The resulting samples were cooled to room temperature. The samples were labeled MSC30-U1 and MSC30-U2, according to the carbon to urea weight ratio.

After urea functionalization, MSC30-U1 and MSC30-U2 were heat-treated over a wide range of temperatures, from 700 to 1300 ºC, in order to vary the degree of functionalization and the contribution of the different nitrogen species. The heat treatment was performed in a tubular furnace with a heating rate of 5 ºC min -1 (or 2.5 ºC min -1 in the case of temperatures above 1100 ºC) under N 2 atmosphere (or Ar for temperatures above 1100 ºC). The obtained samples were labeled MSC30-Y-T, with Y being the functionalization process (U1 or U2) and T being the temperature of the heat treatment (700, 900, 1100, or 1300 ºC).

Physicochemical characterization

The mass contents of carbon, hydrogen, oxygen and nitrogen were determined using a Vario EL cube analyzer (Elementar). 10 mg of each sample was introduced into a combustion vessel and treated at 1700 ºC in a mixed atmosphere of oxygen and helium. The resultant gasses were separated through a chromatographic column and quantified by a thermal conductivity detector with a sensitivity of 40 ppm. In a second separate combustion experiment, the oxygen-containing gases were reduced to CO, from which O quantification was performed.

The surface chemistry and oxidation states of the nitrogen functional groups were studied by X-ray photoelectron spectroscopy (XPS), in a VG-Microtech Multilab 3000 spectrometer, with a non-monochromatised Al K X-ray source under vacuum. The C1s peak at 284.6 eV was used for binding energy corrections. The deconvolution of the N1s spectra was done by least squares fitting using Gaussian-Lorentzian curves.

The morphology and structure of the heat-treated carbon materials were analyzed using a JEM -ARM 200F Cold FEG transmission electron microscopy (TEM) operating at 200 kV. Prior to the experiments, the carbon powder was dispersed in ethanol and deposited in droplets on a holey carbon grid. The chemical composition was followed using energy-dispersive X-ray spectroscopy (EDX), with a JEOL spectrometer (SDD) in scanning transmission microscopy (STEM) mode. STEM images were obtained in high-angle annular dark field (HAADF) mode.

Raman spectroscopy was performed using a Horiba XploRa Raman apparatus equipped with a 50× long-range objective. The spectra were obtained between 800 and 2000 cm -1 , using a holographic grating with 1200 lines per mm and a circularly polarized laser of 638 nm, filtered at 10% of its maximum energy to avoid heating of the samples.

To measure the textural properties of the carbon materials, N 2 and H 2 adsorption isotherms were acquired at -196 ºC, on a 3Flex manometric sorption analyzer (Micromeritics, Atlanta, GA). Prior to adsorption measurements, all samples were outgassed under high vacuum and at 110 ºC for 12h. Pore size distribution (PSD), cumulative pore volume and total surface area (S NLDFT ) were calculated from the combination of N 2 and H 2 adsorption isotherms using a two-dimensional non-local density functional model (2D -NLDFT) available in SAIEUS software (Micromeritics, Atlanta, GA) [START_REF] Jagiello | Carbon slit pore model incorporating surface energetical heterogeneity and geometrical corrugation[END_REF]. H 2 , instead of CO 2 , was used to determine pores between 0.3 and 0.7 nm because H 2 has a smaller cross-sectional area but also a lower quadrupole moment than CO 2 . Therefore, H 2 is more appropriate to characterize carbons with a high heteroatom content [START_REF] Jagiello | Exploiting the adsorption of simple gases O2 and H2 with minimal quadrupole moments for the dual gas characterization of nanoporous carbons using 2D-NLDFT models[END_REF], which is the case in the present study. The other textural characteristics were obtained by integration of the PSDs, such as: total pore volume, V T ; ultramicropore volume (pore diameter below 0.7 nm), V ult-µicro ; supermicropore volume (pore diameter between 0.7 and 2 nm), V sup-µicro ; micropore volume (pore diameter below 2.0 nm), V µicro ; and mesopore volume (pore diameter between 2.0 and 50 nm), V MES . The BET area (A BET ) was calculated by applying the Brunauer-Emmet-Teller theory, using the adequate relative pressure range [START_REF] Rouquerol | Is the BET equation applicable to microporous adsorbents?[END_REF][START_REF] Thommes | Physisorption of gases , with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF].

Electrochemical characterization

The electrochemical ORR tests were performed using a rotating ring-disk electrode (RRDE) in a conventional three-electrode cell on an Autolab PGSTAT302 potentiostat (Metrohm, Netherlands). The RRDE was equipped with a glassy carbon disk (5 mm in diameter) and a platinum ring as the second working electrode. The reference electrode consisted of a reversible hydrogen electrode (RHE) immersed in the working solution and a graphite rod was used as the counter electrode.

The working electrodes for ORR measurements were prepared as follows: 1 mg of carbonbased catalyst was ultrasonically dispersed in 0.25 mL of an aqueous solution of 0.2 wt.% Nafion® and 20 wt.% isopropanol. Then, 33 µL of carbon ink was deposited dropwise on the disk electrode to reach 685 µg•cm -2 of catalyst loading. Linear sweep voltammetry (LSV) was performed from 1.0 to 0.0 V vs RHE in 0.1 M KOH solution saturated with O 2 at a scan rate of 5 mV•s -1 at 1600 rpm. Double-layer capacitive effects was compensated from a previous LSV curve in a N 2 -saturated 0.1 M KOH solution under the same experimental conditions [START_REF] Bouleau | Best practices for ORR performance evaluation of metal-free porous carbon electrocatalysts[END_REF]. The platinum ring was held during the LSV experiments at 1.5 V vs RHE to calculate the H 2 O 2 yield and the number of electrons transferred during the ORR experiments [START_REF] Du | Applications of RDE and RRDE Methods in Oxygen Reduction Reaction[END_REF]. The onset potential (E ONSET ) was determined at a current density of -0.1 mA•cm -2 and the halfwave potential (E 1/2 ) was calculated as the potential at which the current is equal to half of the diffusion current.

Long-term stability tests were performed by chronoamperometric analysis for the most catalytic sample. The working electrode was fixed at 0.70 V vs RHE for 10 000 s at a rotation speed of 1600 rpm and under continuous oxygen bubbling. The resistance to methanol poisoning was also tested by LSV analysis. LSV curves were obtained using the most catalytic sample before and after addition of methanol. For this experiment, methanol was added into the electrolyte to attain a methanol concentration of 1 M.

Results and discussion

Physicochemical properties

A schematic representation of the experimental procedure can be found in Fig. 1a, which recalls what was explained in detail in Section 2.1. This also highlights the sustainability of the process, as no hazardous reagents were used in the synthesis of the N-doped carbon materials. Moreover, urea was used as cheap and abundant nitrogen precursor, which is obtained from ammonia and carbon dioxide. Thus, the proposed synthesis from urea and commercial carbon can be an ideal route to prepare N-doped carbon electrocatalysts while consuming greenhouse gases. High-Resolution Transmission Electron Microscopy (HR-TEM) images (Fig. S1-S2) show that the heat-treated samples possess a disordered structure characteristic of activated carbon materials. Moreover, EDX spectroscopy, combined with STEM in HAADF, demonstrates the homogeneous distribution of oxygen and nitrogen species throughout the carbon structure, proving the successful functionalization of the MSC30 samples after heat treatment (Fig. 1b-1e).

The bulk and surface compositions of the carbon materials determined by elemental analysis and XPS are summarized in Tables S1 andS2, respectively. Table S1 as it is known that temperatures around 550 ºC are necessary to produce in-plane ordering [START_REF] Oanh | Influence of annealing temperature on physical properties and photocatalytic ability of g-C3N4 nanosheets synthesized through urea polymerization in Ar atmosphere[END_REF][START_REF] Liu | Simple pyrolysis of urea into graphitic carbon nitride with recyclable adsorption and photocatalytic activity[END_REF]. According to the literature, HNCO, NH 3 and CO 2 are the three main volatiles formed during urea functionalization at 350 ºC [START_REF] Yang | Effect of thermal program on structure-activity relationship of g-C3N4 prepared by urea pyrolysis and its application for controllable production of g-C3N4[END_REF]. Although HNCO (isocyanic acid) is poisonous, it spontaneously reacts with water and hydrolyzes to carbon dioxide and ammonia. Interestingly, MSC30-U2 exhibits twice the surface nitrogen content (8.9 at.%) than its counterpart synthesized with a MSC30 to urea mass ratio of 1:1 (4.4 at.%). The second heat treatment at increasing temperatures from 700 to 1300 ºC resulted in a progressive removal of nitrogen species, which can be explained by thermal decomposition reactions involving urea-based functional groups [START_REF] Shi | Higher Yield Urea-Derived Polymeric Graphitic Carbon Nitride with Mesoporous Structure and Superior Visible-Light-Responsive Activity[END_REF]. A clear influence of the MSC30 to urea mass ratio on the functionalization degree was observed, i.e., the final nitrogen content was considerably higher when MSC30-U2 was used as starting material compared to MSC30-U1. This is best observed by the surface nitrogen content determined by XPS (Table S2). Total removal of surface nitrogen functionalities was obtained at 1100 °C for MSC30-U1, while a higher temperature, 1300 ºC, was required to obtain a carbon material without surface nitrogen using MSC30-U2 as the starting material. These results demonstrate that the nitrogen content of the final material can be easily tuned by controlling the MSC30 to urea mass ratio.

XPS analyses were performed to evaluate the nitrogen speciation in the carbon materials. The full XPS spectra for all samples can be found in Figs. S3-S4. No significant differences were observed in the high-resolution N1s spectra before carbonization (Fig. S5). Low-temperature treatments in the presence of urea lead to the formation of nitrogen groups on the carbon surface instead of nitrogen atoms embedded in the carbon planes [START_REF] Romero-Cano | Surface functionalization to abate the irreversible capacity of hard carbons derived from grapefruit peels for sodium-ion batteries[END_REF]. Moreover, the formation of C 3 N 4 is ruled out since the N1s spectra are completely different from those of crystalline C 3 N 4 [START_REF] Oanh | Influence of annealing temperature on physical properties and photocatalytic ability of g-C3N4 nanosheets synthesized through urea polymerization in Ar atmosphere[END_REF][START_REF] Liu | Simple pyrolysis of urea into graphitic carbon nitride with recyclable adsorption and photocatalytic activity[END_REF][START_REF] Chidhambaram | Single step transformation of urea into metal-free g-C3N4 nanoflakes for visible light photocatalytic applications[END_REF]. Therefore, the deconvolution of these N1s spectra reveals the presence of amines (-NH 2 ) and imines (=NH) for both MSC30-U1 and MSC30-U2 materials prior to carbonization. The main changes in the surface chemistry of the carbon materials were found after the heat treatments (Fig. 2a and Fig. 2b). After heat treatment in an inert atmosphere, the N1s spectra of the resulting nitrogen-doped carbons were deconvoluted into three groups according to the literature [START_REF] Quílez-Bermejo | Polyaniline-derived N-doped Ordered Mesoporous Carbon Thin Films: Efficient Catalysts towards Oxygen Reduction Reaction[END_REF][START_REF]y a s , . edle s , A. a t o s , The characterization of activated carbons with oxygen and nitrogen surface groups[END_REF][START_REF] Pels | Evolution of nitrogen functionalities in carbonaceous materials during pyrolysis[END_REF]: pyridines at 398.3 eV (blue peak), pyrroles and/or pyridones at 400.3 eV (orange peak), and graphitic nitrogen groups (i.e., quaternary nitrogen) at 401.2 eV (red peak). Amines and imines are converted into heterocyclic functionalities at such high temperatures [START_REF] Pels | Evolution of nitrogen functionalities in carbonaceous materials during pyrolysis[END_REF]. Fig. 2a-b show that the pyrolysis temperature significantly influences the N1s profiles, and consequently, the relative contribution of nitrogen-containing species (see Table S2). Activated carbons treated at the lowest temperature, 700 °C, showed three peaks in the N1s spectrum, which is indicative of a mix of pyridines, pyrroles/pyridones and graphitic N species. Pyridines are the main contribution (around 50%) at 700 ºC, followed by pyrroles/pyridones (≈ 30%) and graphitic-N functionalities (≈ 20%). With increasing temperature, the contribution of pyridines decreases while an increase of graphitic nitrogen is observed, especially for MSC30-U2-1100, in which all nitrogen is in the form of graphitic species (for MSC30-U1-1100, no nitrogen was detected by XPS at this temperature). This is easily observed in the peak quantification, which is summarized for sample MSC30-U2 in Fig. 2c. The gradual elimination of nitrogen functional groups with increasing temperature is evident in Fig. 2d.

No nitrogen content is detected for MSC30-U1-1100, while MSC30-U2-1100 shows an N contribution of 0.6 at.%.

According to the literature, the changes in the N1s profiles are explained by: (i) the greater thermal stability of graphitic nitrogen, and (ii) local changes in the carbon structure, in which pyridine and pyrrole species are transformed into graphitic nitrogen species by condensation reactions at high temperature [START_REF] Pels | Evolution of nitrogen functionalities in carbonaceous materials during pyrolysis[END_REF][START_REF] Sharifi | Formation of active sites for oxygen reduction reactions by transformation of nitrogen functionalities in nitrogen-doped carbon nanotubes[END_REF]. The increase in total graphitic N content, from 0.4 at.% in MSC30-U2-700 to 0.6 at.% in MSC30-U2-1100 can only be explained through the conversion of pyridines and pyrrole/pyridone species into graphitic nitrogen groups. The main feature of this conversion is that it proceeds in the edges of the carbon layers, leading to the formation of edge-type graphitic species, whereas the most common location of graphitic species is the basal plane. Pels et al. proposed that pyridones and pyrrolic N species tend to convert to pyridine species at temperatures above 600 ºC through the loss of oxygen from pyridones and the expansion of pyrrole rings; however, such pyridines are also converted into edge-type graphitic nitrogen species if high temperature treatments are used (Figure 2e) [START_REF] Pels | Evolution of nitrogen functionalities in carbonaceous materials during pyrolysis[END_REF].

This conversion is especially noteworthy since edge-type graphitic nitrogen groups are proposed as the highest ORR catalytic species in N-doped carbon materials [START_REF] Quílez-Bermejo | Metal-free heteroatom-doped carbon-based catalysts for ORR. A critical assessment about the role of heteroatoms[END_REF]. It also worth mentioning that the surface nitrogen species were completely removed from the carbon surface at 1300 ºC, leading to an undoped (or weakly doped) carbon material. These results confirm the selective enrichment of the porous carbon material with graphitic nitrogen species after the 1100 ºC heat treatment of MSC30-U2.

Raman spectroscopy was performed to study the structural order of the heat-treated carbon materials. Figure S6 shows the Raman spectra for the MSC30-U2_T samples. The I D /I G ratio shows a slight variation with heat treatment temperature, indicating that the structural order of carbon materials is progressively changed by the heat treatment. This change is gradual and moderate below 900 ºC, but becomes significant at temperatures above 1100 ºC, for which a clear narrowing of the D and G bands is observed. Nevertheless, these spectra remain typically those of highly disordered and non-graphitizable carbons.

The textural properties of the N-doped and the pristine carbon materials were evaluated using N 2 and H 2 adsorption isotherms, and the results obtained are gathered in Table 1. H 2 adsorption isotherms were performed to characterize the pore size distribution down to 0.3 nm due to smaller cross-section and lower quadrupole moment. Fig. S7 shows the N 2 and H 2 adsorption isotherms and Fig. S8 shows the cumulative pore volumes calculated by 2D-NLDFT. The N 2 adsorption isotherms show that after the functionalization process, part of the porosity of the pristine MSC30 is blocked by the surface groups that hinder the access of the adsorbate to the micropores. Once the heat treatment is applied at low temperature, 700 ºC, these nitrogen groups lead to the production of heterocyclic nitrogen species in the carbon material, which release part of the blocked porosity and, therefore, increase the surface area.

If the highest temperature is used during the heat treatment (i.e., 1300 ºC), a slight loss of surface area is observed due to pore constriction. Both of these effects can be observed in some of the textural properties shown in Table 1 (e.g. A BET , S NLDFT and V µicro ) as well as in the adsorption isotherms (Fig. S7) and the cumulative volume plot (Fig. S8). Fig. S8 shows that most of the pore volume of the pristine MSC30 is in the microporosity range, i.e., corresponding to pores below 2 nm, and which is maintained after urea functionalization and subsequent heat treatments. Similar profiles were found for all materials. Nevertheless, the developed textural properties of MSC30 are still present after functionalization and heat treatments.

Sample Textural Properties

A BET [m 2 /g] V T [cm 3 /g] V MES [cm 3 /g] V µicro [cm 3 /g] V sup-µicro [cm 3 /g] V ult-µicro [cm 3 /g] S NLDFT [m 2 /g] MSC30 3200

Electrocatalytic activity

Figures 3a-b show the LSV curves before and after the heat treatment of the MSC30-U1 and MSC30-U2 samples, respectively. The MSC30 material was also included for comparison (dashed lines). A summary of the ORR electrocatalytic parameters is given in Table S3. Heat treatment of MSC30 in the presence of urea at 350 ºC did not enhance the catalytic activity towards ORR, neither from the point of view of the onset potential, E ONSET , nor from the point of view of the limiting current density, j, as shown by the LSV curves of the MSC30-U1 and MSC30-U2 materials. This fact might be related to the chemical nature of the nitrogen species in these samples, containing mainly amines and imines (see Supporting Information, Fig. S5).

These nitrogen species have been shown to be inactive for oxygen reduction [START_REF] Quílez-Bermejo | Metal-free heteroatom-doped carbon-based catalysts for ORR. A critical assessment about the role of heteroatoms[END_REF]. Moreover, the catalytic activity seems to be slightly reduced due to pore blocking by the non-active nitrogen functional groups, in good agreement with the results obtained by N 2 and H 2 physisorption.

A clear effect of pyrolysis temperature on electrochemical performance was observed. In the case of MSC30-U1 samples, the thermal treatment at 700 ºC improves the limiting current density, compared to pristine MSC30 (dashed line). This increase in catalytic activity continues up to 900 ºC, where the optimum catalytic activity appears to be achieved (onset potential of 0.85 V, E 1/2 = 0.74 V, and limiting current density of -5.5 mA cm -2 ). At temperatures above 900 ºC, the higher the annealing temperature, the lower the current density, with values of -4.9 and -3.5 mA cm -2 for MSC30-U1-1100 and MSC30-U1-1300, respectively. Moreover, a shift to more negative values was observed in the onset potential with increasing temperature. This lower electrochemical performance for the samples treated at 1100 and 1300 ºC is explained by the total removal of surface nitrogen, as evidenced by XPS. These results demonstrate that nitrogen species are necessary to promote highly efficient ORR. For the analogous MSC30-U2 materials (Fig. 3b), the trend is slightly different since the N content is higher at all temperatures. For these materials, the same improvement in catalytic activity is observed up to 900ºC. However, unlike MSC30-U1, there is still an improvement in catalytic activity for the materials heat-treated at 1100 °C. This is noteworthy since the N content of MSC30-U2-1100 (c.a. 0.6 at.%) seems to be high enough to reduce the dioxygen molecules. S3). The presence of this optimized electrocatalytic behavior for oxygen reduction seems to be related to the specific surface chemistry of this sample, containing only graphitic-N species. The following most catalytic samples are MSC30-U1-1100 and MSC30-U1-900 with a significant decrease in E ONSET to 0.85 V, E 1/2 to 0.74 V and number of electrons transferred to 3.6. The samples obtained from MSC30-U1

show lower onset potentials probably due to: i) the lower temperature used during the heat treatment in the case of MSC30-U1-900, which does not allow the complete enrichment in graphitic nitrogen species, or ii) due to the lack of surface nitrogen species at temperatures above 1100 ºC (see the N1s spectrum in Fig. 2a). These results evidence that the most positive onset potentials were obtained for the sample with low nitrogen content (0.6 at.%), but in the form of exclusively graphitic nitrogen species.

The hydrogen peroxide yield, which is indicative of ORR mechanisms, is shown in Fig. 3d and 3e. Interestingly, all materials show low H 2 O 2 yield at higher potentials, meaning that the number of electrons transferred is close to 4 in the useful potential range (0.6-1.0 V vs. RHE), a crucial parameter for the energy density of fuel cell stations. Among all the materials, those prepared using MSC30-U2 were the ones that showed lower H 2 O 2 yield and, consequently, higher number of electrons transferred during ORR, especially the material treated at 1100 ºC.

Fig. 3f shows the number of electrons transferred and hydrogen peroxide yield for the most catalytic sample, MSC30-U2-1100, across the potential range. In addition, a minimum H 2 O 2 yield is observed, about 25%, and the number of electrons transferred remains constant, 3.5, at the less positive potentials. Therefore, the highly porous and graphitic nitrogen-enriched carbon material, obtained by urea functionalization and heat treatment at 1100 ºC, was found to be a highly competitive catalyst for ORR with high selectivity towards the 4-electron pathway.

To support the role of graphitic N species in the catalytic activity of the MSC30 samples, we added two materials for comparison purposes in Fig. 4a. The first material is MSC30-1100, i.e., the sample MSC30 treated at 1100 ºC, to ensure that the enhancement of catalytic activity material is Pt/C, i.e., the commercial platinum-based catalyst. Fig. 4a-b reveal that the highly efficient 4-electron ORR catalysis of MSC30-U2-1100 is related to the graphitic nitrogen species since the sample treated without nitrogen functionalization shows significantly lower catalytic activity. This finding further confirms that selective graphitic enrichment by simple N doping with urea and high-temperature treatment is an efficient and low-cost method to produce excellent metal-free N-doped carbon catalysts for ORR. Graphitic nitrogen species indeed modify the electron density of the adjacent carbon atoms, promoting ORR, as already concluded by theoretical and experimental studies [START_REF] Quílez-Bermejo | On the origin of the effect of pH in ORR for non-doped and edge-type quaternary Ndoped metal-free carbon-based catalysts[END_REF][START_REF] Quílez-Bermejo | Towards understanding the active sites for the ORR in N-doped carbon materials through fine-tuning of nitrogen functionalities: an experimental and computational approach[END_REF][START_REF] Sharifi | Formation of active sites for oxygen reduction reactions by transformation of nitrogen functionalities in nitrogen-doped carbon nanotubes[END_REF][START_REF] Ikeda | Carbon Alloy Catalysts: Active Sites for Oxygen reduction Reaction[END_REF][START_REF] Bao | A first-principles study of the role of quaternary-N doping on the oxygen reduction reaction activity and selectivity of graphene edge sites[END_REF]. Methanol was added to the electrolyte to reach a concentration of 1 M, and the ORR experiments were recorded before and after MeOH addition. Fig. S9 shows that MeOH oxidation prevails over ORR when using Pt-based catalyst. At the same time, carbon materials, and in particular MSC30-U2-1100, are catalysts that show excellent tolerance to MeOH poisoning. Fig. 4d shows the LSV curves of MSC30-U2-1100 before and after methanol addition. The catalytic performance of MSC30-U2-1100 showed no differences in E ONSET or limiting current density under the action of methanol, supporting the strong resistance of the graphitic N-enriched carbon material to methanol poisoning.

Comparison with the literature

The paramount role of graphitic N was confirmed by a comparative analysis of different published carbon-based catalysts for the oxygen reduction reaction (ORR) in alkaline electrolyte. Fig. 4e and Table S4 show a detailed literature search of N-doped carbon materials with the majority of the nitrogen (> 50 at.%) in the form of graphitic or pyridinic N.

As observed, metal-free carbon materials with pyridines as the main type of nitrogen show better catalytic activity compared to undoped carbon materials. The undoped carbon catalysts show a sluggish ORR activity with a 2-electron mechanism, while pyridine-based carbon materials show the reduction of oxygen molecules through an intermediate number of electrons transferred but a similar E ONSET . However, the carbon materials with the majority of the nitrogen as graphitic species show the best catalytic performance for ORR in the N-doped carbon materials, with E ONSET around 0.9 V vs RHE and a number of electrons transferred close to 4. This confirms that graphitic N appears to be the most promising nitrogen species to boost the ORR catalytic activity of N-doped carbon materials and that very good ORR catalysts can be obtained by selectively enriching activated carbons with graphitic N, as presented in this study. Graphitic N, especially that located at the edges of the carbon layers, Finally, Fig. 4f and Table S5 show the best performance of metal-free carbon-based catalysts in the recent literature. The green circle of this figure reflects the catalytic behavior of the MSC30-U2-1100 material. As can be seen, despite the superb ORR performance of this sample, there are carbon-based materials that still exhibit superior catalytic activities.

However, it is important to note that these high performances were achieved by doping with multiple heteroatoms, using expensive reagents, such as graphene, or tedious and complex methodologies. Moreover, all these highly catalytic carbons were obtained from N-containing precursors, which make impossible to control the textural properties of the resulting materials.

In contrast, this study proposed a selective and green post-functionalization route to obtain highly catalytic carbons without disturbing significantly the textural properties of the pristine materials, making it a promising synthesis method not only for preparing highly catalytic carbons for ORR, but also for improving the properties of carbon materials in multiple applications.

Conclusions

Highly porous and graphitic N-enriched carbon materials were prepared by nitrogen doping with urea of a commercial activated carbon material, MSC30, using a simple postfunctionalization method. Urea and MSC30 were mixed and heat-treated to obtain N-doped carbon materials with different nitrogen contents and functional groups distributions. Since the heat treatment had no significant effect on the textural properties, the functionalized carbon materials thus prepared corroborated the key role of graphitic nitrogen in the successful catalysis of the oxygen reduction reaction (ORR). Moreover, these materials showed great potential in terms of catalytic activity (E ONSET = 0.88 V vs RHE; H 2 O 2 selectivity < 20%; j = -5.0 mA•cm -2 ), exhibiting catalytic performance similar to that of leading metal-free catalysts and even commercial Pt-based electrocatalysts. Furthermore, this study provided an efficient and low-cost method to produce graphitic N-enriched carbon with excellent catalytic activity for ORR, high stability and resistance to methanol poisoning while retaining most of the textural properties intact.

Fig. 1 :

 1 Fig. 1: (a) Schematic representation of the experimental procedure. Commercial activated carbon, MSC30, was functionalized with a MSC30 to urea mass ratios of 1:1 or 1:2. These samples were heat-treated from 700 to 1300 ºC in a nitrogen (˂ 1100 °C) or argon (≥ 1100 °C) atmosphere, with T being the temperature of the heat treatment. (b) HAADF image of MSC30-U2-1100 and EDX analyses of (c) carbon, (d) nitrogen and (e) oxygen of the same image.

  yield of the materials. Heat treatment of the pristine MSC30 in the presence of urea at 350 ºC successfully introduced nitrogen into the bulk and at the surface of the activated carbon. Treatment with urea at moderate temperatures often leads to the formation of C 3 N 4 . However, the temperature of 350 ºC does not allow the formation of C 3 N 4 from urea,

Fig. 2 :

 2 Fig. 2: N1s spectra of (a) MSC30-U1, and (b) MSC30-U2 samples after heat treatment at 700, 900, 1100 and 1300 ºC. The blue peak at 398.3 eV represents pyridines, the orange peak at 400.3 eV represents pyrroles or pyridones, and the red peak at 401.2 eV represents graphitic nitrogen species. (c) Contributions of each kind of nitrogen species in the MSC30-U2 samples.

Fig. 3 :

 3 Fig. 3: LSV curves for N-doped carbon materials obtained after heat treatment of (a) MSC30-U1, and (b) MSC30-U2 in 0.1 M KOH solution saturated with O 2 at a scan rate of 5 mV•s -1 . (c) Comparison of ORR curves for the most active materials in the same conditions as in (a) and (b). (d,e) H 2 O 2 yield calculated from the current of the platinum ring during ORR experiments. (f) H 2 O 2 yield and number of electrons transferred for MSC30-U2-1100.

Fig. 4 :

 4 Fig. 4: (a) LSV curves, and (b) H 2 O 2 yield of pristine MSC30, MSC30-1100, and MSC30-U2-1100 in an O 2 -saturated 0.1 M KOH solution at a scan rate of 5 mV•s -1 . A 20 wt.% Pt/C catalyst has been included in these figures following the same procedure. (c) Chronoamperometric analysis of MSC30-U2-1100 and Pt/C catalyst at 0.70 V vs RHE in an O 2 -saturated 0.1 M KOH solution. (d) LSV curves of sample MSC30-U2-1100 before and after the addition of 1 M methanol into an O 2saturated 0.1 M KOH solution at a scan rate of 5 mV•s -1 . (e) Bibliographic search of metal-free carbon materials with the majority of nitrogen in the form of graphitic or pyridinic N; undoped carbon materials were included for comparison. (f) Bibliographic search of the latest published metal-free carbon-based catalysts towards ORR, including the commercial Pt/C catalysts for comparison; the data were obtained from[START_REF] Yang | Carbon-Based Metal-Free ORR Electrocatalysts for Fuel Cells : Past, Present, and Future[END_REF][START_REF] Gabe | Modeling of oxygen reduction reaction in porous carbon materials in alkaline medium. Effect of microporosity[END_REF][START_REF] Quílez-Bermejo | Oxygen reduction catalysis of Ndoped carbons prepared via heat treatment of polyaniline at over 1100 o C[END_REF][START_REF] Maldonado | Influence of nitrogen doping on oxygen reduction electrocatalysis at carbon nanofiber electrodes[END_REF][START_REF] Quílez-Bermejo | Towards understanding the active sites for the ORR in N-doped carbon materials through fine-tuning of nitrogen functionalities: an experimental and computational approach[END_REF].

  produces a dissociative ORR mechanism through bridging (C-O-O-C) bonding modes, leading to the chemisorption of the two oxygen atoms from dioxygen molecules and, consequently, to O-O bond cleavage and a highly selective 4-electron pathway [26]. At the same time, pyridine N and undoped carbons chemisorb oxygen molecules through a terminal (C-O-O) bonding mode, leading to an associative mechanism in which the 2-electron pathway predominates.

Table 1 :

 1 Textural properties of the nitrogen-doped carbon materials obtained from N 2 and H 2 adsorption isotherms. N 2 adsorption isotherms were used to calculate A BET , and the combination of N 2 and H 2 adsorption isotherms were used to calculate all other parameters.

Acknowledgements:

The French research team thanks ANR-15-IDEX-04-LUE and the TALiSMAN project (2019-000215), financed by the European Regional Development Fund (ERDF). The authors from UA thank MICINN and ERDF (project RTI2018-095291-B-I00) for financial support.


H Hiesinger
email: hiesinger@uni-muenster.de

C H Van Der Bogert

A Wedler

R Jaumann

U Mall

J W Head

M Anand

B Jolliff

P Pinet

L Xiao

M Ivanov

B Gundlach

N Schmitz

M Schweitzer

C Bergemann-Mecucci

A Jaime

M C Desanctis

S Besse

M Massironi

E Hauber

A Frigeri

R Tartese

F Rull

F Mcdonald

J Carpenter

The Rima Bode Region -Absolute Model Ages of a Candidate Future Lunar Landing Site

whether they are published or not. The documents may come

Introduction:

The Rima Bode region offers unique opportunities to study lunar volcanism as it hosts several compositionally different mare units, as well as pyroclastic deposits [e.g., 1,2]. In addition, the area is affected by ray material from Copernicus crater, ejecta from Eratosthenes crater, and exhibits highland material and ejecta from the Imbrium basin [e.g., 1,2]. In short, the Rima Bode site is extremely geologically complex and permits the study of several key science questions concerning early crustal differentiation, volcanic history and evolution, impact history, and the potential for utilizing pyroclastic deposits for the extraction of volatiles and other potential lunar resources (i.e., ISRU).

Last year, we presented a first evaluation of the Rima Bode region as a potential landing site for a future lander mission [2]. Since then, ESA has formed a topical team of lunar scientists to investigate the scientific potential of several candidate landing sites, including Ina D, Copernicus and Tsiolkovsky craters, and a long traverse across Mare Imbrium, to name only a few. Among the discussed landing sites, Rima Bode ranked very high, because of its broad scientific potential and the relatively low technical risks. All geologically diverse units can be accessed with a rover traverse less than ~150-200 km long and slopes less than ~7°. ESA is currently developing plans to utilize the European Large Lunar Lander (EL3) for science-driven missions. EL3 will allow us to bring at least 1.5 tons of payload to the lunar surface and a first mission without a rover to a polar landing site is scheduled for 2029. This mission will be followed by more complex and comprehensive missions that will include a rover in the 2030's. Thus, we envision a rover that will be able to travel 10s -100s of kilometers, have night survival capabilities, and carry a comprehensive suite of instruments (e.g., LIBS, Raman, IR-spectrometer, multispectral stereo cameras) to characterize and pre-select interesting samples. A continuously operating ground-penetrating radar would yield information on the sub-surface regolith structure along rover paths [3].

The lander will also carry instruments, including a geophysical package (e.g., seismometer, heat flow probe, laser retroreflector, etc.) that acquires long term stable measurements and could perform geochemical analyses of samples (e.g., in situ radiometric age determination with a laser ablation resonance ionization mass spectrometer [4]). ISRU technology demonstrations are also planned for the lander. With our mission concept, we aim to comprehensively study lunar volcanism with respect to timing, spatial distribution, composition, mineralogy, eruption style, volumes, and ISRU potential.

In a companion abstract [5], we present a preliminary geologic map of the area, which was created in order to further investigate the local geology for the selection of the "best" landing site to accomplish the mission goals. One major question concerns the detailed stratigraphy of the mapped geologic units and their absolute model ages (AMA). Ages of volcanic deposits, determined via crater size-frequency distributions (CSFD) and/or radiometric and exposure ages, allow the reconstruction of the thermal evolution of the Moon [e.g., 6,7].

Data and Method: We used Kaguya and LROC images to map count areas and perform CSFD measurements. The Kaguya images have a pixel scale of about 10 m on the surface and large incidence angles favorable for CSFD measurements. The images were imported into ArcGIS. Within ArcGIS, we used CraterTools [8] to perform our CSFD measurements. The CSFDs were then plotted with CraterStats [9], using the lunar production and chronology functions of [10]. The technique of CSFD measurements has been described extensively [e.g., [10][11][12][13][14]. For our crater counts, we tried to map homogeneous areas on several geologic units and particularly paid attention to avoid obvious secondary craters.

Results: With our CSFD measurements we determined AMAs for several geologic units around the Rima Bode landing site south of Bode C crater (11.6°N / 4.7°W) (Fig. 1). In particular, we dated three dark presumably pyroclastic deposits, three mare units, the ejecta blanket of crater Bode E, and also selected a count area on the ray of Copernicus crater, which crosses the landing area (Fig. 1). Our CSFD of this ray material (CR1 in Fig. 1) yields an AMA of ~3.45 Ga, which is much older than the canonical age of Copernicus crater of ~800 Ma [e.g., 15 and references therein]. We interpret this old age to reflect the age of the underlying mare basalt, which is covered by only a thin veneer of ray material that did not reset the crater statistics. The mare unit M1 immediately north of the ray shows an age of ~3.70 Ga with a resurfacing at ~2.54 Ga. The second mare unit M2 farther north is topographically lower, is younger, and has an AMA of ~2.96 Ga. Our CSFDs for the third and southernmost mare unit M3 yield an AMA of ~3.27 Ga. This mare unit was dated by combining two adjacent count areas. The counted craters on the ejecta blanket of Bode E (C1) are in equilibrium up to diameters of ~800 m. Although we were unable to accurately date the crater, we can infer an age of at least ~3.8 Ga. Because Bode E is superposed on the ejecta material of the Imbrium basin (i.e., Fra Mauro Formation) it is plausible that Bode E is younger than ~3.92 Ga, which we assume as the age of Imbrium. The dark mantle deposit D1 of unit Idp2 in the geologic map of [5] shows an AMA of ~3.71 Ga and, thus, is older than the adjacent mare basalts. This stratigraphic sequence is supported by the sharp contact between the pyroclastic deposits and the mare basalts, which is consistent with a superposition of the mare basalts on top of the pyroclastic deposits. The pyroclastic deposits D2 of unit Ir [5] exhibit a wide range of AMAs with evidence for at least one resurfacing event. In particular, our CSFD measurement indicates an old AMA of ~4.07 Ga and a resurfacing at ~3.52 Ga. We interpret the old age to represent the age of underlying highlands or potential cryptomare and the younger age to be related to the deposition of the pyroclastic material. Finally, for the dark mantle deposits D3 of unit Idp1 [5], we find an extremely young AMA (~510 Ma), which we consider unrealistic in relation to the relative stratigraphy and absolute model ages of the other units. This CSFD measurement will need to be re-investigated.

2169.pdf 53rd Lunar and Planetary Science Conference (2022)

Conclusions: On the basis of our landing site assessment, we consider the region south of crater Bode C to be safe and to be scientifically very interesting particularly for a better understanding of volcanism on the Moon. With a capable lander/rover combination delivered to the lunar surface by EL3, major scientific advances could be achieved and various techniques of ISRU could be tested in preparation for a human lunar outpost. The new preliminary results provide first order information on the stratigraphy of some of the major geologic units in the vicinity of the landing site that can be used to develop a more detailed mission scenario, to guide a rover, and to design a sampling strategy.

Fig. 1

 1 Fig. 1 Rima Bode landing site (yellow star) and count areas (CR1: Copernicus ray; M1-3: Mare units; D1-3: Dark mantle deposits; C1: Bode E ejecta


Hartman-Watson Law

N Demni
email: demni@ccr.jussieu.fr

LAGUERRE PROCESS AND GENERALIZED

Keywords: Generalized Hartman-Watson law, Gross-Richards formula, Laguerre process, Special functions of matrix argument

In this paper, we study complex Wishart processes or the so-called Laguerre processes (Xt) t≥0 . We are interested in the behaviour of the eigenvalue process, we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman-Watson law as well as the law of T0 := inf{t, det(Xt) = 0} when the size of the matrix is 2.

 established the following result : p F q (a 1 , . . . , a p , b 1 , . . . , b q ; X) = det(x n-j i p F q (a 1 -j + 1, . . . , a p -j + 1, . . . , b q -j + 1; x i ))

V (X)

Introduction

The Real Wishart process is a symmetric matrix-valued process which was introduced by M.F. Bru (1989) as follows : Let B t = (B ij (t)) i,j be a n × m Brownian matrix and define X t = B T t B t . The process (X t ) t≥0 satisfies the following stochastic differential equation (SDE)

dX t = B T t dB t + dB T t B t + nI m dt = X t dN t + dN T t X t + nI m dt, X 0 = B T 0 B 0
where I m denotes the unit matrix, the superscript T stands for the transpose, √ X t is the matrix square root of the positive matrix X t and (N t ) t≥0 is a m × m Brownian matrix. This process is called the Wishart process of dimension n, of size m, starting from X 0 and is denoted W (n, m, X 0 ). Then, the W (δ, m, X 0 ) where δ runs over the Gindikin ensemble {1, . . . , m -1}∪]m -1, ∞[ is defined as the unique solution of the latter SDE with δ instead of n. Thus, it can be viewed as an extension of the squared Bessel process to higher dimension. In this way, [START_REF] Donati-Martin | Some properties of Wishart process and a matrix extension of the Hartman-Watson law[END_REF] tried to extend some well known properties of the squared Bessel processes to the matrix case and derived expressions such as the Laplace transform and the tail distribution of some random variables, in which many multivariate special functions of symmetric matrix argument appear, such as Gamma, modified Bessel and hypergeometric functions [START_REF] Muirhead | Aspects of Multivariate Statistical Theory[END_REF]. However, the latter is quite complicated to deal with and to our knowledge, there are no more precise results on the law of these variables. Nevertheless, in the complex case, hypergeometric functions of Hermitian matrix argument can be expressed as a determinant of a matrix Date: November 28, 2006. where X is a m × m Hermitian matrix, (x i ) are its eigenvalues, p F q denotes the standard hypergeometric functions with scalar argument, V (X) = i<j (x i -x j ) is the Vandermonde determinant and p F q is the hypergeometric function with Hermitian matrix argument defined by : p F q (a 1 , . . . , a p , b 1 , . . . , b q ; X)

= k≥0 τ (a 1 ) τ • • • (a p ) τ (b 1 ) τ • • • (b q ) τ C τ (X) k!
where τ = (τ 1 , . . . , τ m ) is a partition of length ≤ m and of weight k (i.e.

τ 1 ≥ τ 2 ≥ • • • ≥ τ m , i τ i = k), (
a) τ is the generalized Pochammer symbol and C τ is the so-called zonal polynomial. We refer to [START_REF] Macdonald | Symmetric Functions and Hall Polynomials[END_REF] for further details and Lassalle (1991Lassalle ( ), (1991a) ) for analogous expressions for multivariate orthogonal polynomials. The determinantal representation above is due to the fact that the zonal polynomial is identified with the (normalized) Schur functions defined by : s τ (x 1 , . . . , x m ) = det(x

τ j +m-j i ) det(x m-j i )
Consequently, one can use integral representations as well as other properties of standard hypergeometric functions to get, at least when m = 2, some results that are till now unknown in the Wishart case. The rest of this paper consists of seven sections, which are respectively devoted to the following topics: in section 2, we introduce the Laguerre process of integer dimension. In section 3, we study the behaviour of the eigenvalue process. Then, in section 4, we define the Laguerre process of positive real dimension. Section 5 is devoted to the absolute-continuity relations, from which we deduce the Laplace transform of the so-called generalized Hartman-Watson law as well as the tail distribution of T 0 , the first hitting time of 0. In section 6, we focus on the case m = 2 for which we invert this Laplace transform, and finally, in section 7, we compute the density of S 0 := 1/(2T 0 ).

Laguerre Process of integer index

Let B be a n×m complex Brownian matrix starting from B 0 , i.e. B = (B ij ) where the entries B ij are independent complex Brownian motions, so we can write B = B 1 + iB 2 where B 1 , B 2 are two independent real Brownian matrices. We are interested in the matrix-valued process X t := B ⋆ t B t which satisfies the following SDE :

dX t = dB ⋆ t B t + B ⋆ t dB t + 2nI m dt (1)
Definition 1. (X t ) t≥0 is called the Laguerre process of size m, of dimension n and starting from X 0 = B ⋆ 0 B 0 , and will be denoted by L(n, m, X 0 ). Remark 1. For m = 1, (X t ) t≥0 is a squared Bessel process of dimension 2n, denoted by BESQ(2n, X 0 ).

Remark 2. Set X t = (X ij (t)) i,j . One can easily check that

dX ii (t) = 2 X ii (t)dγ i (t) + 2ndt 1 ≤ i ≤ m,
where (γ i ) 1≤i≤m are independent Brownian motions, thus, X ii is a BESQ(2n, (X 0 ) ii ).

Remark 3. The equation above implies that :

d(tr(X t )) = 2 tr(X t )dβ t + 2nmdt (2)
where β is a Brownian motion. Consequently, (tr(X t )) t≥0 is a BESQ(2nm, tr(X 0 )) of dimension 2nm starting from tr(X 0 ). One can also deduce from (1) that for every i, j, k, l ∈ {1, • • • , m}:

dX ij , dX kl t = 2(X il δ kj + X kj δ il )dt,
which differs from equation (I-1-5) derived by Bru (1989) since for a complex Brownian motion γ, we have d γ, γ t = 0 and γ, γ t = 2t .

2.1. Infinitesimal generator. Let H m , H+ m be respectively the space of m × m Hermitian matrices and the space of m × m positive definite Hermitian matrices. On the space of Hermitian matrix-argument functions, we define the matrix-valued differential operators :

∂ ∂x := ∂ ∂x jk j,k , ∂ ∂y := ∂ ∂y jk j,k , ∂ ∂z := ∂ ∂x jk -i ∂ ∂y jk j,k ,
We also define:

∂ ∂z 2 ij := k ∂ 2 ∂z ik ∂z kj , ∂ ∂x ∂ ∂y ij := k ∂ 2 ∂x ik ∂y kj Proposition 1. Let functions f satisfying: ∂f ∂x ij = ∂f ∂x ji , ∂f ∂y ij = - ∂f ∂y ji for all i, j.
Then, the infinitesimal generator of a Laguerre process L(n, m, x) is given by:

L = 2n tr(ℜ ∂ ∂z ) + 2[tr(xℜ ∂ ∂z 2 ) + tr(yℑ ∂ ∂z 2 )] (3)
where ∂ ∂z is the operator defined above.

Remark 4. Using the fact that x T = x, y T = -y and tr(AB) = tr(BA) = tr(B T A T ) for any two matrices A and B, we can see that

tr y ∂ ∂y ∂ ∂x = tr ∂ ∂x ∂ ∂y y = tr y ∂ ∂x ∂ ∂y ⇒ yℑ( ∂ ∂z ) 2 = 2tr y ∂ ∂x ∂ ∂y

Eigenvalues of Laguerre Process

In this section, we will suppose that n ≥ m. The following result is due to König and O'Connell (2001), [START_REF] Katori | Symmetry of matrix-valued processes and noncolliding diffusion particle systems[END_REF] and Bru in the real case (1989a) :

Theorem 1. Let λ 1 (t), • • • , λ m (t)
denote the eigenvalues of X t . Suppose that at time t = 0, all the eigenvalues are distinct. Then, the eigenvalue process (λ 1 (t), . . . , λ m (t)) satisfies the following stochastic differential system:

dλ i (t) = 2 λ i (t) dβ i (t) + 2   n + k =i λ i (t) + λ k (t) λ i (t) -λ k (t)   dt 1 ≤ i ≤ m, t < τ,
where the (β i ) 1≤i≤m are independent Brownian motions and τ is defined by τ := inf{t, λ i (t) = λ j (t) for some (i, j)}.

Remark 5. With the help of the SDE satisfied by the eigenvalues, we can compute the ones satisfied by both processes (tr(X)) and (det(X)): the former is done. For the second, we find that for t < T 0 := inf{t, det(X t ) = 0} and for r ∈ R:

d(det(X t )) = 2 det(X t ) tr(X -1 t )dν t + 2(n -m + 1) det(X t ) tr(X -1 t )dt d(log(det(X t )) = 2 tr(X -1 t )dν t + 2(n -m) tr(X -1 t )dt, d(det(X t ) r ) = 2r(det(X t )) r tr(X -1 t )dν t + 2r(n -m + r)(det(X t )) r tr(X -1 t
)dt so, we can see that for n = m, log(det(X) is a local martingale and so is (det(X)) m-n . Lemma 1. Take X 0 ∈ H+ m . Then for n ≥ m, X t ∈ H+ m . Proof : In fact, this result is a direct consequence of the fact that for n = m, log det(X) is a local martingale, and so is (det(X)) m-n . Hence, for n ≥ m, these two continuous processes tend to infinity when t → T 0 which is possible only if T 0 = ∞, because every continuous local martingale is a time-changed Brownian motion.

Corollary 1. If λ 1 (0) > • • • > λ m (0)
, then, the process U defined by

U (t) = 1 i<j (λ i (t) -λ j (t))
, t < τ is a local martingale.

Proof : We could follow the proof given by Bru (1989a) or make straightforward computations using the derivatives of the Vandermonde function. But we prefer use a result from König and O'Connell (2001): for n ≥ m, the eigenvalue process is the Vtransform (in the Doob sense) of the process obtained from m independent BESQ(2(nm + 1)). Thus, if G and Ĝ denote respectively the infinitesimal generators of these two processes, then, G(h) = 0 and, for all

C 2 function f , Ĝ(f ) = 1 V G(V f ) ⇒ Ĝ(U ) = 1 V G(1) = 0.
Corollary 2. If at time t = 0, the eigenvalues of X are distinct, then, they will never collide, i.e. τ = ∞ almost surely.

Proof : This result follows from the fact that the continuous process U tends to infinity when t → τ which is possible only if τ = ∞ almost surely (We use the same argument as before).

3.1. Additivity Property. The proof of this result is similar to the one derived by Bru in the real case (1989): Proposition 2. If (X t ) t≥0 and (Y t ) t≥0 are two independent Laguerre processes L(n, m, X 0 ) and L(p, m, Y 0 ) respectively, then the process (X t + Y t ) t≥0 is a Laguerre process L(n + p, m, X 0 + Y 0 ). Now, we introduce the Laguerre processes of noninteger dimensions δ.

Laguerre Processes with noninteger dimensions

Let X be a Laguerre process L(n, m, X 0 ) with n ≥ m. If X 0 ∈ H+ m , and if √ X t stands for the symmetric matrix square root of X t , it is easy to show that the matrix O defined by O

t := √ X t -1 B ⋆ t , where X t = B ⋆ t B t , satisfies O ⋆ O = OO ⋆ = I m . Thus, dγ t = O t dB t = X t
-1 B ⋆ t dB t is a m × m complex Brownian matrix. Replacing this expression in (1), one obtains :

dX t = X t dγ t + dγ ⋆ t X t + 2nI m dt Theorem 2. If (B t
) is a m × m complex Brownian matrix, then for every X 0 ∈ H+ m and for all δ ≥ m, the SDE

dX t = X t dB t + dB ⋆ t X t + 2δI m dt (4)
has a unique strong solution in H+ m . Furthermore, if the eigenvalues are distinct at time t = 0, then they satisfy the stochastic differential system:

dλ i (t) = 2 λ i (t) dβ i (t) + 2   δ + k =i λ i (t) + λ k (t) λ i (t) -λ k (t)   dt 1 ≤ i ≤ m,
where the (β i ) 1≤i≤m are independent Brownian motions.

Proof : The proof of the second part of the theorem is the same as before with δ instead of n. So, we have to prove the first part. Note first that (det(X t )), (log det(X t )) and (det(X t ) r ) verify the same SDE with δ instead of n. Hence, arguing as before, we can see that T 0 = ∞ almost surely. On the other hand, the map a → a 1/2 is analytic in H+ m (see Roger and Williams 1987, p. 134), so, the SDE has a unique strong solution for all t ≥ 0.

Definition 2. Such a process is called the Laguerre process of dimension δ, size m and initial state X 0 . It will be denoted by L(δ, m, X 0 ). Remark 6. Any process (X t ) t≥0 solution of ( 4) is a diffusion whose infinitesimal generator is given by:

L = 2δ tr(ℜ( ∂ ∂z )) + 2[tr(xℜ( ∂ ∂z ) 2 ) + tr(yℑ( ∂ ∂z ) 2 )] Remark 7. A simple computation shows that d X ij , X kl t = 2(X il (t)δ kj + X kj (t)δ il )dt, for all i, j, k, l ∈ {1, . . . , m}
Now, we focus on both existence and uniqueness when δ > m -1 and X 0 ∈ H + m (see Bru (1989) for the real case).

4.1. The Process X + . If X is a Hermitian matrix, let X + be the Hermitian matrix max(X, 0). If we denote by (λ i ) the eigenvalues of X, then (λ

+ i = max(λ i , 0)) are those of X + .
Theorem 3. For all δ ∈ R + and X 0 = x ∈ H m , the stochastic differential equation

dX t = X + t dB t + dB ⋆ t X + t + 2δI m dt (5)
has a solution in H m .

Proof : The mapping a → √ a + is continuous on H m . Hence, X exists up to its explosion time (Ikeda and Watanabe 1989, theorem. 2. 3). Furthermore, from

|| √ X + || 2 + ||δI || 2 ≤ δ 2 + ||X|| 2 ≤ C(1 + ||X|| 2 ),
we can deduce that this explosion time is infinite almost surely (Ikeda and Watanabe 1989, theorem 2. 4).

Proposition 3. If λ 1 (0) > . . . > λ m (0) ≥ 0, then, for all t < S := inf{t, λ i = λ j for some (i, j)}, the eigenvalues of X + verify the following differential system:

dλ i (t) = 2 λ + i (t)dν i (t) + 2   δ + k =i λ + i (t) + λ + k (t) λ i (t) -λ k (t)   dt, 1 ≤ i ≤ m,
Proof : This differential system can be shown in the same way as in theorem 1 using :

dX ij , dX kl t = 2(X + il (t)δ kj + X + kj (t)δ il )dt, for all i, j, k, l ∈ {1, . . . , m} Proposition 4. If λ 1 (0) > . . . > λ m (0) ≥ 0, then, for all δ > m -1, t > 0, λ m (t) ≥ 0 .
Proof : First, we note that S = ∞ almost surely. Indeed, one can easily show that the process U defined by :

U (λ 1 (t), . . . , λ m (t)) = 1 i<j (λ i (t) -λ j (t)
) is a local martingale. For the proof, we follow in the same way as Bru (1989).

Theorem 4. If λ 1 (0) > . . . > λ m (0) ≥ 0, then, for all δ > m -1, (4) has a unique solution in H +
m in the sense of probability law. Proof : By Proposition 4, the solution of the SDE (5) remains positive for all t > 0, thus, it is a solution of (4).

Theorem 5. Let H + m be the space of positive Hermitian matrices. Then, whenever the SDE (4) has a solution in H + m , for fixed t, its distribution is given by its Laplace transform:

E X 0 (exp -(tr uX t )) = (det(I m + 2tu)) -δ exp(-tr(X 0 (I m + 2tu) -1 u)), (6)
for all u in H + m . Proof : For s ∈ H + m , let g(t, s) = ∆ -δ t exp(-V (t, s))
, where

∆ t = det(I m + 2ut), W t = (I m + 2ut) -1 u, V (t, s) = tr(sW t ),
First, note that W ∈ H m . To proceed, we need a lemma :

Lemma 2. The function g satisfies the heat equation: ∂g ∂t = L g where L is the infinitesimal generator of X.

Proof of the lemma : If we write s = x + iy, then, using the fact that x is symmetric, y is skew-symmetric and W is Hermitian, we can see that tr(sW t ) = tr(xM + iyN ) where

M = W + W 2 N = W -W 2 .
Observing that M T = M and N T = -N , we can deduce that g satisfies the conditions of Proposition 1. Besides,

∂g ∂t = -g(2δ tr(W t ) -2 tr(sW 2 t )) tr(y ∂ 2 g ∂x∂y + ∂ 2 g ∂y∂x ) = -ig tr(yW 2 ), tr(x ∂ 2 g ∂x 2 - ∂ 2 g ∂y 2 ) = g tr(x(M 2 + N 2 )) = g tr(xW 2 ).
Finally, noting that tr(M ) = tr(W ), we obtain the equality. Now, we consider the process (Z(t, X t )) defined by Z(t, X t ) = g(t 1 -t, X t ) for all t ≤ t 1 for fixed t 1 . From the lemma, we deduce that Z is a bounded local martingale and thus is a martingale. So, the result follows from a simple application of the optional stopping theorem.

Corollary 3. Let (X t ) t≥0 be a Laguerre process L(δ, m, x) where x ∈ Hm + . For δ > m -1, its semi-group is given by the following density:

p δ t (x, y) = 1 (2t) mδ Γ m (δ) exp -( 1 2t tr(x + y)) (det y) δ-m 0 F 1 (δ; xy 4t 2 )1 {y>0}
with respect to Lebesgue measure dy = p≤q dy 1 pq p<q dy 2 pq where y = y 1 + iy 2 and 0 F 1 is a hypergeometric function of Hermitian matrix argument (Chikuze 1976, Gross and[START_REF] Gross | Special functions of matrix argument[END_REF].

Proof : In fact, this result can be easily deduced from the case where δ = n is integer, since, in this case, X t is a non-central complex Wishart variable W (n, 2tI m , x) [START_REF] James | Distributions of matrix variates and latent roots derived from normal samples[END_REF] with density given by:

f t (x, y) = 1 (2t) mn Γ m (n) exp -( 1 2t tr(x + y)) (det y) n-m 0 F 1 (n; xy 4t 2 )1 {y>0}
with respect to dy. Hence, taking δ instead of n and denoting by W t this new variable (starting from x), we can see that : (we will use |y| to denote det(y))

E x (e -tr uWt ) = 1 (2t) mδ Γ m (δ) e -trx 2t y>0 exp - 1 2t tr((I + 2ut)y) |y| δ-m 0 F 1 (δ; xy 4t 2 )dy = 2t mδ |x| -δ Γ m (δ) e -trx 2t z>0 exp(-2t tr(x -1 2 (I + 2ut)x -1 2 z)|z| δ-m 0 F 1 (δ; z)dz = exp(- trx 2t )|I + 2ut| -δ exp tr( x 2t (I + 2ut) -1 ) = |I + 2ut| -δ exp - 1 2t tr(x(I + 2ut) -1 (I + 2ut -I )) = |I + 2ut| -δ exp -tr(x(I + 2ut) -1 u)
which is equal to (6).

Remark 8. In the last proof, we used the change of variables z = x 1/2 yx 1/2 which gives dz = |x| m dy. For the second integral, see Faraut and Korànyi (1994), proposition XV.1.3.

Remark 9. The expression of the semi-group extends continuously to the degenerate case, namely:

p δ t (0 m , y) = 1 (2t) mδ Γ m (δ) exp -( tr(y) 2t ) (det y) δ-m 1 {y>0}
where 0 m denotes the null matrix.

Corollary 4. For δ > m -1, the semi-group of eigenvalue process is given by:

q t (x, y) = V (y) V (x) det 1 2t y j x i ν/2 e - (x i +y j ) 2t I ν ( √ x i y j t )
where x = (x 1 , . . . , x m ), y = (y 1 , . . . , y m ) so that x 1 > . . . > x m > 0, y 1 > . . . > y m > 0, δ = m + ν such that ν > -1 and I ν denotes the modified Bessel function [START_REF] Lebedev | Special Functions And Their Applications[END_REF].

Proof : The expression of the semi-group can be computed using Karlin and MacGregor formula (1959) since, for δ > m -1, the eigenvalue process is the h-transform of the process consisted of m independent BESQ(2(δ -m + 1)) conditioned never to collide, as stated by König and O'Connell (2001). Another proof is given by Péché (2003, p. 68).

Here, we will deduce the expression of q t (x, y) from p t (x, y) following [START_REF] Muirhead | Aspects of Multivariate Statistical Theory[END_REF], namely, by projection on the unitary group : we will use Weyl integration formula, then give a determinantal representation of hypergeometric functions of two matrix arguments. First, we state Weyl integration formula [START_REF] Faraut | Infinite dimensional harmonic analysis and probability[END_REF]) in the complex case: for any Borel function f ,

Hm f (A)dA = C m U (m) R m f (uau * )α(du)(V (a)) 2 da 1 . . . da m , where C m = π m(m-1) Γ m (m) , U (
m) is the unitary group, α is the normalized Haar measure on U (m), a = diag(a i ) and A = uau * . Hence, the semi-group of the eigenvalue process is given by [START_REF] James | Distributions of matrix variates and latent roots derived from normal samples[END_REF]):

q t (x, y) = C m (V (y) 2 ) U (m) p t (x, uỹu * )α(du) = C m (V (y) 2 ) (2t) mδ Γ m (δ) m i,j=1 e - x i +y j 2t m i=1 y j δ-m U (m) 0 F 1 (δ; xuỹu * 4t 2 )α(du) = π m(m-1) (V (y) 2 ) (2t) m(m+ν) Γ m (m)Γ m (m + ν) m i,j=1 e - x i +y j 2t m i=1 y j ν 0 F 1 (m + ν;
x 4t 2 ; ỹ), where ỹ = diag(y j ) , x is a positive definite matrix whose eigenvalues are x 1 , . . . , x m , 0 F 1 in the third line is an hypergeometric function with two matrix arguments [START_REF] Gross | Special functions of matrix argument[END_REF] and δ = m + ν, ν > -1. Next, we need a lemma.

Lemma 3. Let B, C ∈ H m and let (b i ), (c i ) be respectively their eigenvalues. Then,

p F q ((m + µ i ) 1≤i≤p ,(m + φ j ) 1≤j≤q ; B, C) = π m(m-1) 2 (p-q-1) Γ m (m) p i=1 (Γ(µ i + 1)) m Γ m (m + µ i ) q j=1 Γ m (m + φ j ) (Γ(φ j + 1)) m det ( p F q ((µ i + 1) 1≤i≤p , (1 + φ j ) 1≤j≤q ; b l c f ) l,f V (B)V (C) for all µ i , φ j > -1, 1 ≤ i ≤ p, 1 ≤ j ≤ q.
Proof : Recall that the hypergeometric function of two matrix arguments is given by the following series:

p F q ((a i ) 1≤i≤p , (e j ) 1≤j≤q ; B, C) = ∞ k=0 τ p i=1 (a i ) τ q j=1 (e j ) τ C τ (B)C τ (C) C τ (I)k! , It is well known that C τ (B) = k!d τ (m) τ s τ (b 1 , . . . , b m ),
where s τ is the Schur function and d τ = s τ (1, . . . , 1) is the representation trace or degree [START_REF] Gross | Special functions of matrix argument[END_REF]Richards 1991, Faraut 2006). Substituting in the series, one gets:

p F q ((m + µ i ) 1≤i≤p , (m + φ j ) 1≤j≤q ; B, C) = ∞ k=0 τ p i=1 (m + µ i ) τ q j=1 (m + φ j ) τ s τ (B)s τ (C) (m) τ ,
Now, we write:

(m + µ i ) τ = m r=1 Γ(µ i + m + k r -r + 1) Γ(µ i + m -r + 1) = m r=1 Γ(µ i + 1 + k r + δ r ) Γ(µ i + m -r + 1) = π m(m-1)/2 (Γ(µ i + 1)) m Γ m (m + µ i ) m r=1 (µ i + 1) kr+δr , δ r = m -r
Doing the same for each (m + φ j ) τ and for (m) τ , we can see that: where β = m(m -1) 2 (p -q -1). To get the desired result, we use the "Hua formula" [START_REF] Faraut | Infinite dimensional harmonic analysis and probability[END_REF]) :

p F q ((m + µ i ) 1≤i≤p , (m + φ j ) 1≤j≤q ;B, C) = π β Γ m (m) p i=1 (Γ(µ i + 1)) m Γ m (m + µ i ) q j=1 (Γ(φ j + 1)) m Γ m (m + φ j ) ∞ k=0
Lemma 4. Given an entire function f, i.e. f (z) = ∞ k=0 e k z k , then det(f (b i c j )) i,j V (B)V (C) = ∞ k=0 τ m r=1 e kr+δr s τ (B)s τ (C) s τ (I m ) .
Thus, we get:

p F q ((m + µ i ) 1≤i≤p ,(m + φ j ) 1≤j≤q ; B, C) = π m(m-1) 2 (p-q-1) Γ m (m) p i=1 Γ(µ i + 1) Γ m (m + µ i ) q j=1 Γ m (m + φ j ) Γ(φ j + 1) det ∞ k=0 p i=1 (µ i +1) k q j=1 (φ j +1) k (b l cp) k k! l,p V (B)V (C) = π m(m-1) 2 (p-q-1) Γ m (m) p i=1 Γ(µ i + 1) Γ m (m + µ i ) q j=1 Γ m (m + φ j ) Γ(φ j + 1) det ( p F q ((µ i + 1) 1≤i≤p , (1 + φ j ) 1≤j≤q ; b l c f ) l,f V (B)V (C)
For p = 0 and q ≥ 1, it reads :

0 F q ((m + φ j ) 1≤j≤q ; B, C) = π -m(m-1) 2 (q+1) Γ m (m) q j=1 Γ m (m + φ j ) (Γ(φ j + 1)) m det ( 0 F q ((1 + φ j ) 1≤j≤q ; b l c f ) l,f V (B)V (C) ,
and similarly,

0 F 0 (B, C) = Γ m (m) π m(m-1) 2
det(e b l c f ) l,f V (B)V (C) which can be viewed as Harish-Chandra formula for the "Itzykson-Zuber" integral [START_REF] Collins | Intégrales Matricielles et Probabilités Non-commutatives[END_REF]. We now proceed to the end of the proof. Taking p = 0, q = 1, B = x 4t 2 , C = ỹ, we get:

0 F 1 (m + ν; x 4t 2 ; ỹ) = (4t 2 ) m(m-1)/2 Γ m (m + ν)Γ m (m) π m(m-1) (Γ(ν + 1)) m det 0 F 1 ((ν + 1); x i y j 4t 2 ) V (x)V (y)
The expression of q t (x, y) follows from a simple computation and from the fact that:

0 F 1 ((ν + 1); x i y j /4t 2 )) i,j Γ(ν + 1) = 2t √ x i y j ν I ν ( √ x i y j t )
Proposition 5. The measure defined by ρ(dx) = (det(x)) δ-m dx on Hm + is invariant under the semi-group, i. e, ρP t = ρ.

Proof: Denote by P t the semi-group of Laguerre process L(δ, m, x) for δ > m -1. Then, we have to show that

x>0 P t f (x)ρ(dx) = y>0 f (y)ρ(dy) f ∈ C 0 ( Hm + ).
This follows by a similar computation and the same arguments as in the proof of corollary 3.

Remark 10. For Wishart processes, it is easy to see that µ(dx) := (det(x))

δ 2 -m+1
2 1 {x>0} dx is invariant under the semi-group.

Girsanov Formula and Absolute-continuity Relations

The index ν > -1 of a L(δ, m, x) is defined by ν = δ -m. In this section, we will discuss in the same way as in [] to derive absolute-continuity relations between different indices.

5.1. Positive Indices. Take a matrix-valued Hermitian predictable process H. Let Q δ x be the probability law of L(δ, m, x) for δ > m -1 and x ∈ H+ m . Define:

L t = t 0 tr(H s dB s + H s dB s ) 2 , Φ t = exp (L t - 1 2 t 0 tr(H 2 s )ds),
where B is a complex Brownian matrix under Q δ x . We can easily see that the process β defined by β t = B t -t 0 H s ds is a Brownian matrix under the probability

P H x | Ft := Φ t • Q δ x | Ft , Furthermore, (X t ) t≥0 is a solution of dX t = X t dβ t + dβ ⋆ t X t + ( X t H t + H t X t + 2δI m )dt. ( 7
) For H t = ν √ X t -1 , (7) becomes dX t = X t dβ t + dβ ⋆ t X t + 2(δ + ν)I m dt, so that (X t ) t≥0 is a L(δ + ν, m, x) under P H
x . Thus, we proved that : Theorem 6. For δ > m -1,

Q δ+ν x | Ft = exp ν 2 t 0 tr( X s -1 dB s + X s -1 dB s ) - ν 2 2 t 0 tr(X s -1 )ds • Q δ x | Ft . (8) Proposition 6. Q m+ν x | Ft = det(X t ) det(x) ν/2 exp - ν 2 2 t 0 tr(X s -1 )ds • Q m x | Ft . (9)
Proof : We know that ∇ u (det(u)) = det(u)u -1 , hence, ∇ u (log(det(u))) = u -1 . Then, using the fact that for δ = m, (log(det(X t ))) is a local martingale, we get from Itô formula that:

log(det(X t )) = log(det(X 0 )) + t 0 tr(X s -1 ( X s dB s + dB ⋆ s X s )) = log(det(X 0 )) + t 0 tr( X s -1 dB s + X s -1 dB s ).
From ( 9), it follows that:

Corollary 5. Q m x (exp - ν 2 2 t 0 tr(X -1 s )ds |X t = y) = det(y) det(x) -ν/2 p m+ν t (x, y) p m t (x, y) = Γ m (m) Γ m (m + ν) (det(z)) ν/2 0 F 1 (m + ν, z) 0 F 1 (m, z) := Ĩν (z) Ĩ0 (z) ,
where z = xy 4t 2 . Now, we state the following asymptotic result:

Corollary 6. Let X be a Laguerre process L(m, m, x), then, as t → ∞:

4 (m log t) 2 t 0 tr(X s ) -1 ds L → T 1 (β)
where T 1 is the first hitting time of 1 by a standard Brownian motion β.

Proof : From (9), we deduce that:

Q m x (exp - 2ν 2 (m log t) 2 t 0 tr(X -1 s )ds |X t = ty) = Γ m (m) Γ m (m + 2ν/m log t) (det(xy/4t)) ν/m log t 0 F 1 (m + 2ν/m log t, xy/4t 2 ) 0 F 1 (m, xy/4t 2 )
.

Noting that (t m ) -ν/m log t = e -ν , and since both hypergeometric functions converge to 1 as t → ∞, we obtain:

Q m x exp(- 2ν 2 (m log t) 2 t 0 tr(X -1 s )ds|X t = ty t→∞ -→ e -ν
Then, since

lim t→∞ t m 2 p m t (x, 2y) = lim t→∞ e -tr(x)/2t Γ m (m) e -tr(y) 0 F 1 (m, xy 2t ) = e -tr(y)
Γ m (m) we get:

Q m x (exp - 2ν 2 (m log t) 2 t 0 tr(X -1 s )ds ) = y>0 Q m x exp(- 2ν 2 (m log t) 2 t 0 tr(X -1 s )ds|X t = y p m t (x, y)dy = y>0 Q m x exp(- 2ν 2 (m log t) 2 t 0 tr(X -1 s )ds|X t = ty t m 2 p m t (x, ty)dy t→∞ -→ e -ν ,
by dominated convergence Theorem.

5.2. Negative Indices. Take 0 < a ≤ det(x). The same computation as in parag. 5.1 with

H t = -ν √ X t -1 , 0 < ν < 1, shows that Q m-ν x | F t∧Ta = det(x) det(X t∧Ta ) ν/2 exp - ν 2 2 t∧Ta 0 tr(X s -1 )ds Q m x | F t∧Ta
where T a := inf{t, det(X t ) = a}. Letting a → 0 and using the fact that T 0 = ∞ a.s under Q m x , we get :

Q m-ν x | F t∧T 0 = det(x) det(X t ) ν/2 exp ν 2 2 t 0 tr(X s -1 )ds Q m x | Ft = det(x) det(X t ) ν Q m+ν x | Ft
Proposition 7. For all t > 0 and 0 < ν < 1,

Q m-ν x (T 0 > t) = Γ m (m) Γ m (m + ν) det( x 2t ) ν 1 F 1 (ν, m + ν, - x 2t )
Proof : From the absolute-continuity relation above, we deduce that :

Q m-ν x (T 0 > t) = Q m+ν x det(x) det(X t ) ν ,
On the other hand, using the expression of the semi-group, one has :

Q δ x (det(X t ) s ) = (2t) ms Γ m (s + δ) Γ m (δ) 1 F 1 (-s; δ; - x 2t ) = (2t) ms Γ m (s + δ) Γ m (δ) exp(-tr( x 2t )) 1 F 1 (δ + s; δ; x 2t )
by Kummer relation (cf Th 7. 4. 3 in Muirhead 1982). Taking s = -ν, we are done.

Generalized Hartman-Watson law

Henceforth, we will write F to denote one-dimensional hypergeometric functions. We define the generalized Hartman-Watson law as the law of

t 0 tr(X -1 s )ds under Q m x (•|X t = y).
Its Laplace transform is given by:

Q m x (exp -ν 2 2 t 0 tr(X -1 s )ds |X t = y) = Γ m (m) Γ m (m + ν) det(z) ν/2 0 F 1 (m + ν, z) 0 F 1 (m, z) (10)
z = xy/4t 2 . Recall that for m = 1, this is the well-known Hartman-Watson law and that its density was computed by [START_REF] Yor | Loi de l'indice du lacet Brownien et distribution de Hartman-Watson[END_REF]. Here, we will investigate the case m = 2. The Gross and Richards formula is written for p = 0 and q = 1 :

0 F 1 (m + ν, z) = det(z m-j i 0 F 1 (m + ν -j + 1, z i )) V (z) ,
where (z i ) denote the eigenvalues of z and

V (z) = i<j (z i -z j ) is the Vandermonde determinant . Noting that Γ m (m + ν) = m j=1 Γ(m + ν -j + 1), then : (10) = det(z (m-j)/2 i I m+ν-j (2 √ z i )) det(z (m-j)/2 i I m-j (2 √ z i ))
Without loss of generality, we will take t = 1.

Proposition 8. For m = 2, let λ 1 > λ 2 be the eigenvalues of √ xy. Then, the density of the generalized Hartman-Watson law is given by :

f (v) = √ λ 1 λ 2 v pπ √ 2πv 3 1 0 ∞ 0 z sinh(p √ 1 -z 2 )e -2 √ λ 1 λ 2 z cosh y e -2(y 2 -π 2 ) v (sinh y) sin( 4πy v )dzdy 1 0 1 0 u cosh(pu √ 1-x 2 ) √ 1-x 2 I 0 (2 √ λ 1 λ 2 ux )dudx
, for v > 0, where p = λ 1 -λ 2 . Furthermore, if λ 1 = λ 2 := λ, then:

f (v) = 4λve 2π 2 v π 2 √ 2πv 3 ∞ 0 g(y)e -2y 2 v (sinh y) sin( 4πy v )dy 1 F 2 ( 1 2 ; 1; 2; λ 2 )
, where

g(y) = 1 3 + π 2 I 2 (2λ cosh y) + L 2 (2λ cosh y) 2λ cosh y ,
and L 2 is the Struve function [START_REF] Gradshteyn | Table of integrals, series and products[END_REF].

Proof : For m = 2 , (10) becomes:

(10) = λ 1 I ν+1 (λ 1 )I ν (λ 2 ) -λ 2 I ν+1 (λ 2 )I ν (λ 1 ) λ 1 I 1 (λ 1 )I 0 (λ 2 ) -λ 2 I 1 (λ 2 )I 0 (λ 1
) , so, using the integral representations below (Brychkov, Marichev, Prudnikov 1986, p. 46) : and (Gradshteyn and Ryzhik 1994, p. 734):

x(aI ν+1 (ax )I ν (bx ) -bI ν+1 (bx )I ν (ax )) = (a 2 -b 2 ) x 0 uI ν (au)I ν (bu)du with x = 1, a = λ 1 , b = λ 2 ,
π 2 I ν ( a 2 ( b 2 + c 2 + b))I ν ( a 2 ( b 2 + c 2 -b)) = a 0 cosh(b √ a 2 -x 2 ) √ a 2 -x 2 I 2 ν (cx )dx where a > 0, ℜ(ν) > -1, with a = 1, b = (λ 1 -λ 2 )u := pu et c = 2 √ λ 1 λ 2 u
, the numerator of ( 10) is then equal to:

2 π (λ 2 1 -λ 2 2 ) 1 0 1 0 u cosh(pu √ 1 -x 2 ) √ 1 -x 2 I 2 ν (2 λ 1 λ 2 ux )dudx .
Taking ν = 0, the denominator is then equal to:

2 π (λ 2 1 -λ 2 2 ) 1 0 1 0 u cosh(pu √ 1 -x 2 ) √ 1 -x 2 I 0 (2 λ 1 λ 2 ux )dudx .
Thus, (10) becomes:

1 0 1 0 u cosh(pu √ 1-x 2 ) √ 1-x 2 I 2 ν (2 √ λ 1 λ 2 ux )dudx 1 0 1 0 u cosh(pu √ 1-x 2 ) √ 1-x 2 I 0 (2 √ λ 1 λ 2 ux )dudx
Now, we only have to use the integral representation of I 2 ν [START_REF] Yor | Loi de l'indice du lacet Brownien et distribution de Hartman-Watson[END_REF]:

I 2 ν (2 λ 1 λ 2 ux ) = 1 2iπ C e 2 √ λ 1 λ 2 ux cosh ω e -2νω dω = 1 2iπ C e 2 √ λ 1 λ 2 ux cosh ω ∞ 0 2ωe -vν 2 /2 (2πv 3 ) 1/2 e -2ω 2 v dvdω
where C is the contour indicated in [START_REF] Yor | Loi de l'indice du lacet Brownien et distribution de Hartman-Watson[END_REF], hence, the density function is given by:

f (v) = 1 iπ √ 2πv 3 1 0 1 0 C uω cosh(pu √ 1-x 2 ) √ 1-x 2 e 2 √ λ 1 λ 2 ux cosh ω e -2ω 2 v dudxdω 1 0 1 0 u cosh(pu √ 1-x 2 ) √ 1-x 2 I 0 (2 √ λ 1 λ 2 ux )dudx 1 {v>0}
We can simplify this expression by integrating over C to see that the numerator is equal to [START_REF] Yor | Loi de l'indice du lacet Brownien et distribution de Hartman-Watson[END_REF]:

√ λ 1 λ 2 v π √ 2πv 3 1 0 1 0 ∞ 0 u 2 x cosh(pu √ 1 -x 2 ) √ 1 -x 2 e -2 √ λ 1 λ 2 ux cosh y e --2(y 2 -π 2 ) v (sinh y) sin( 4πy v )dudxdy Setting z = ux, The numerator is written √ λ 1 λ 2 v π √ 2πv 3 1 0 u 0 ∞ 0 z u cosh(p √ u 2 -z 2 ) √ u 2 -z 2 e -2 √ λ 1 λ 2 z cosh y e -2(y 2 -π 2 ) v (sinh y) sin( 4πy v )dudzdy,
that we can integrate with respect to u to get

√ λ 1 λ 2 v pπ √ 2πv 3 1 0 ∞ 0 z sinh(p 1 -z 2 )e -2 √ λ 1 λ 2 z cosh y e -2(y 2 -π 2 ) v (sinh y) sin( 4πy v )dzdy.
Now, we prove the second part. In this case, p = 0 and we have to evaluate :

λve 2π 2 v π √ 2πv 3 1 0 1 0 ∞ 0 u 2 x √ 1-x 2 e -2λux cosh y e -2y 2 v (sinh y) sin( 4πy v )dudxdy 1 0 1 0 uI 0 (2 λux ) √ 1-x 2 dudx
Setting z = ux, the numerator reads :

λve 2π 2 v π √ 2πv 3 1 0 ∞ 0 z 1 -z 2 e -2λz cosh y e -2y 2 v (shy)sin( 4πy v )dzdy,
Integration with respect to z yields (Gradshteyn and Ryzhik 1994, p. 369):

λve 2π 2 v π √ 2πv 3 ∞ 0 g(y)e -2y 2 v (shy)sin( 4πy v )dy
For the denominator, we use the fact that d dz (zI 1 (z )) = zI 0 (z ), which yields:

1 0 1 0 uI 0 (2 λux ) √ 1 -x 2 dudx = 1 0 I 1 (2 λx ) 2λx √ 1 -x 2 dx
Then, the following formula

a 0 x α-1 (a 2 -x 2 ) β-1 I ν (cx )dx = 2 -ν-1 a 2β+α+ν-2 c ν Γ(β)Γ((α + ν)/2) Γ(β + (α + ν)/2)Γ(ν + 1) (11) 1 F 2 ( α + ν 2 ; β + α + ν 2 ; ν + 1; a 2 c 2 4 )
taken with α = 0, a = 1, β = 1/2, c = 2λ, ν = 1 gives:

1 0 I 1 (2 λx ) 2λx √ 1 -x 2 dx = π 4 1 F 2 ( 1 2 ; 1; 2; λ 2 )
We can proceed differently : let λ 1 = λ 2 + h then (10) reads:

((λ 2 + h)I ν+1 (λ 2 + h)I ν (λ 2 ) -λ 2 I ν+1 (λ 2 )I ν (λ 2 + h))/h ((λ 2 + h)I 1 (λ 2 + h)I 0 (λ 2 ) -λ 2 I 1 (λ 2 )I 0 (λ 2 + h))/h .
Next, we let h → 0. As usual, we first compute the numerator and then take ν = 0. To do this, we shall evaluate :

A = lim h→0 (λ 2 + h)I ν+1 (λ 2 + h) -λ 2 I ν+1 (λ 2 ) h B = lim h→0 I ν (λ 2 + h) -I ν (λ 2 ) h which are equal respectively to d dx (xI ν+1 (x)) and d dx (I ν (x)) taken for x = λ = λ 1 = λ 2 .
Using the differentiation formula d dx (x ν I ν (x)) = x ν I ν-1 (x) (Lebedev 1972, p. 110), we get:

d dx (xI ν+1 (x)) = -νI ν+1 (x) + xI ν (x), d dx (I ν (x)) = - ν x I ν (x) + I ν-1 (x),
thus:

N = I ν (λ)(-νI ν+1 (λ) + λI ν (λ)) -λI ν+1 (λ)(- ν λ I ν (λ) + I ν-1 (λ)) = λ(I ν 2 (λ) -I ν+1 (λ)I ν-1 (λ)) A = I ν 2 (λ) -I ν+1 (λ)I ν-1 (λ) I 0 2 (λ) -I 1 (λ)I -1 (λ)
Using the integral representation below (Gradshteyn and Ryzhik 1994, p. 757):

I µ (z)I ν (z) = 2 π π/2 0 cos((µ -ν)θ)I µ+ν (2z cos θ)dθ, ℜ(µ + ν) > -1.
the numerator is written as : Thus,using (11), the denominator is equal to

N = 2 π π/2 0 (1 -cos 2θ)I 2 ν (2λ cos θ)dθ = 4 π π/2 0 (sin 2 θ)I 2 ν (2λ cos θ)dθ = 4 π 1 0 1 -r 2 I 2 ν (2λr)dr,
D = 4 π 1 0 1 -r 2 I 0 (2λr)dr = π 4 1 F 2 ( 1 2 ; 1; 2; λ 2 )
Finally, the integral representation of I ν gives :

f (u) = λue 2π 2 /u π √ 2πu 3 ∞ 0 e -2y 2 /u sinh(y) sin 4πy u 1 0 r √ 1 -r 2 e -2λr cosh y dr du 1 0 √ 1 -r 2 I 0 (2λr)dr = λue 2π 2 /u π √ 2πu 3 ∞ 0 g(y)e -2y 2 /u sinh(y) sin 4πy u du 1 0 √ 1 -r 2 I 0 (2λr)dr , 7. The Law of T 0 Recall that: For 0 < ν < 1, Q m-ν x (T 0 > t) = Γ m (m) Γ m (m + ν) det( x 2t ) ν 1 F 1 (ν, m + ν, - x 2t )
Proposition 9. Let m = 2 and λ 1 > λ 2 be the eigenvalues of x. The density of

S 0 := 1/(2T 0 ) under Q m-ν
x is given by:

f (u) = (λ 1 λ 2 ) ν u 2ν-2 e -(λ 1 +λ 2 )u Γ(ν + 1)Γ(ν) 1 F 1 (2, ν + 1, λ 1 u) -1 F 1 (2, ν + 1, λ 2 u) (λ 1 -λ 2 )
Corollary 7. If λ 1 = λ 2 := λ, the density is written:

f (u) = 2λ 2ν u 2ν-1 e -λu Γ(ν + 2)Γ(ν) 1 F 1 (ν -1, ν + 2, -λu) Proof : Recall first that when m = 1, S 0 L = γ ν /x
, where γ ν is a Gamma variable with density r ν-1 e -r dr. With the help of the Gross-Richards formula, it follows that for m = 2,

Q m-ν x (S 0 ≤ u) = (λ 1 λ 2 ) ν (λ 1 -λ 2 )Γ 2 (ν + 2) u 2ν (λ 11 F 1 (ν, ν + 2, -λ 1 u) 1 F 1 (ν -1, ν + 1, -λ 2 u) -λ 21 F 1 (ν, ν + 2, -λ 2 u) 1 F 1 (ν -1, ν + 1, -λ 1 u)),
where S 0 := 1/(2T 0 ). This is a C ∞ function in u. Hence, we will compute its derivative to get the density. Recall that :

d dz 1 F 1 (a, b, z) = a b 1 F 1 (a + 1, b + 1, z), thus : f (u) = d du Q m-ν x (S 0 ≤ u) = K(ν, λ 1 , λ 2 )u 2ν-1 (A -B)
where

K(ν, λ 1 , λ 2 ) = (λ 1 λ 2 ) ν Γ 2 (ν + 2)(λ 1 -λ 2 ) A = 2ν((λ 11 F 1 (ν, ν + 2, -λ 1 u) 1 F 1 (ν -1, ν + 1, -λ 2 u) -λ 21 F 1 (ν, ν + 2, -λ 2 u) 1 F 1 (ν -1, ν + 1, -λ 1 u)) B = ν ν + 2 ((λ 2 1 u 1 F 1 (ν + 1, ν + 3, -λ 1 u) 1 F 1 (ν -1, ν + 1, -λ 2 u) -λ 2 2 u 1 F 1 (ν + 1, ν + 3, -λ 2 u) 1 F 1 (ν -1, ν + 1, -λ 1 u))
. Then, we use the contiguous relation :

b 1 F 1 (a, b, z) -b 1 F 1 (a -1, b, z) = z 1 F 1 (a, b + 1, z) to see that λ 1 u 1 F 1 (ν + 1, ν + 3, -λ 1 u) = (ν + 2)( 1 F 1 (ν, ν + 2, -λ 1 u) -1 F 1 (ν + 1, ν + 2, -λ 1 u)) λ 2 u 1 F 1 (ν + 1, ν + 3, -λ 2 u) = (ν + 2)( 1 F 1 (ν, ν + 2, -λ 2 u) -1 F 1 (ν + 1, ν + 2, -λ 2 u)) implies that: f (u) = K 1 (ν, λ 1 , λ 2 )u 2ν-1 (C + D -E -F )
where

K 1 (ν, λ 1 , λ 2 ) = ν(λ 1 λ 2 ) ν Γ 2 (ν + 2)(λ 1 -λ 2 ) C = λ 11 F 1 (ν, ν + 2, -λ 1 u) 1 F 1 (ν -1, ν + 1, -λ 2 u) D = λ 11 F 1 (ν + 1, ν + 2, -λ 1 u) 1 F 1 (ν -1, ν + 1, -λ 2 u) E = λ 21 F 1 (ν, ν + 2, -λ 2 u) 1 F 1 (ν -1, ν + 1, -λ 1 u) F = λ 21 F 1 (ν + 1, ν + 2, -λ 2 u) 1 F 1 (ν -1, ν + 1, -λ 1 u),
Applying again the above contiguous relation yields:

λ 1 u 1 F 1 (ν + 1, ν + 2, -λ 1 u) = (ν + 1)( 1 F 1 (ν, ν + 1, -λ 1 u) -1 F 1 (ν + 1, ν + 1, -λ 1 u)) λ 2 u 1 F 1 (ν + 1, ν + 2, -λ 2 u) = (ν + 1)( 1 F 1 (ν, ν + 1, -λ 2 u) -1 F 1 (ν + 1, ν + 1, -λ 2 u)) λ 2 u 1 F 1 (ν, ν + 2, -λ 2 u) = (ν + 1)( 1 F 1 (ν -1, ν + 1, -λ 2 u) -1 F 1 (ν, ν + 1, -λ 2 u)) λ 1 u 1 F 1 (ν, ν + 2, -λ 1 u) = (ν + 1)( 1 F 1 (ν -1, ν + 1, -λ 1 u) -1 F 1 (ν, ν + 1, -λ 1 u))
Replacing in the expression of f , we obtain

f (u) = K 2 (ν, λ 1 , λ 2 )u 2ν-2 (G -H), where K 2 (ν, λ 1 , λ 2 ) = ν(ν + 1)(λ 1 λ 2 ) ν Γ 2 (ν + 2)(λ 1 -λ 2 ) G = 1 F 1 (ν + 1, ν + 1, -λ 2 u) 1 F 1 (ν -1, ν + 1, -λ 1 u) H = 1 F 1 (ν + 1, ν + 1, -λ 1 u) 1 F 1 (ν -1, ν + 1, -λ 2 u)
Eventually, writing Γ 2 (ν + 2) = Γ(ν + 2)Γ(ν + 1) = ν(ν + 1)Γ(ν + 1)Γ(ν)

1 F 1 (a, a, z) = e -z 1 F 1 (a, b, -z) = e -z 1 F 1 (b -a, b, z), we get

f (u) = (λ 1 λ 2 ) ν u 2ν-2 e -(λ 1 +λ 2 )u Γ(ν + 1)Γ(ν) 1 F 1 (2, ν + 1, λ 1 u) -1 F 1 (2, ν + 1, λ 2 u) (λ 1 -λ 2 )
Th case λ 1 = λ 2 is treated in the same way as before (for the Hartman-Watson law). In fact , writing λ 1 = λ 2 + h and letting h → 0, we see that the density is given by : f (u) = λ 2ν u 2ν-2 e -2λu Γ(ν + 1)Γ(ν)

d dλ 1 F 1 (2, ν + 1, λu) = 2λ 2ν u 2ν-1 e -2λu Γ(ν + 2)Γ(ν) 1 F 1 (3, ν + 2, λu) = 2λ 2ν u 2ν-1 e -λu Γ(ν + 2)Γ(ν) 1 F 1 (ν -1, ν + 2, -λu)

Conclusion

The Gross-Richards formula has been the main ingredient in this paper, since it enables us to express more explicitly the special functions of matrix argument. The case m = 3 can be treated in the same way, but computation becomes too complicated. So, if we want to deal with the general case, it will be convenient to find a more explicit formula. Indeed, Schur functions can be expressed as polynomials in the elementary symmetric functions e r or as polynomials in the completely symmetric functions h r . More precisely, we have :

s λ = det(e λ i -i+j ) 1 ≤ i, j ≤ n s λ = det(h λ ′ i -i+j ) 1 ≤ i, j ≤ n
where λ is a partition of length ≤ n, and λ ′ is the conjugate of λ [START_REF] Macdonald | Symmetric Functions and Hall Polynomials[END_REF]. So, using these two identities, can we improve our results?

9. Appendix: special functions 9.1. The hypergeometric series. The multivariate hypergeometric functions were studied by [START_REF] Muirhead | Aspects of Multivariate Statistical Theory[END_REF] in the real symmetric case, [START_REF] Chikuze | Partial differential equations[END_REF] for the complex Hermitian case and Faraut and Korànyi (1994) in a more general setting. For Hermitian matrix argument, they are defined by: p F q ((a i ) 1≤i≤p , (b j ) 1≤j≤q ; X) = and C τ (X) is the zonal polynomial of X such that :

(tr(X)

) k = τ ⊥k C τ (X)
Several normalizations for this polynomial exist in the litterature but we consider this one. This polynomial is symmetric, homogeneous, of degree k in the eigenvalues of X and is an eigenfunction of the following differential operator :

∆ X = m i=1 x 2 i ∂ 2 ∂x 2 i + 2 m i=1 1≤k =i≤m x 2 i x i -x k ∂ ∂x i
Besides, it is identified with the Schur function s τ and C τ (Y X) = C τ ( √ Y X √ Y ) for any Hermitian matrix Y . It is well-known that, if p = q + 1, then the hypergeometric series is convergent for 0 ≤ ||X|| < 1 (|| • || is the norm given by the spectral radius) , if p ≤ q, then it converges everywhere and else, it diverges. 9.2. The modified Bessel function [START_REF] Lebedev | Special Functions And Their Applications[END_REF]. The modified Bessel function with index ν ∈ R is given by the following series :

I ν (z) = ∞ k=0 1 k!Γ(ν + k + 1) z 2 2k+ν , z ∈ C.
It can be represented through standard hypergeometric functions 0 F 1 and 1 F 1 :

I ν (z) = 1 Γ(ν + 1) z 2 ν 0 F 1 (ν + 1; z 2 )

  ) τ • • • (a p ) τ (b 1 ) τ • • • (b q ) τ C τ (X) k!where τ = (k 1 , . . . , k m ) is a partition of weight k and length m such that k 1 ≥ . . . ≥ k m , (a) τ is the generalised Pochammer symbol defined by:(a) τ = m i=1 Γ(a + k i -i + 1) Γ(a -i + 1) , τ = (k 1 , . . . , k m )

Acknowledgments I would like to thank C. Donati-Martin (my Ph. D. supervisor), J. Faraut and M. Yor for helpful comments and encouragements.