Enantio-enriched CPL-active helicene-bipyridine-rhenium complexes

The incorporation of a rhenium atom within an extended helical -conjugated bi-pyridine system impacts the chiroptical and photophysical properties of the resulting neutral or cationic complexes, leading to the first examples of rhenium-based phosphors that exhibit circularly polarized luminescence.

2,2'-Bipyridine (bipy) derivatives are widely used N,N'bidentate ligands in coordination chemistry, giving access to a great variety of complexes. 1 The luminescence properties of d 6 transition metal polypyridyl complexes have been increasingly studied for the development of new metal-based luminescent materials and sensing probes. 2 Among them, [Re(N,N')(CO) 3 X] 0/+ complexes (X = halide, pyridyl (py) or isocyanide (CNR)) exhibit room-temperature (RT) phosphorescence from triplet metal-toligand (ML) and/or intraligand charge-transfer (ILCT) states. [START_REF] Ak | [END_REF]4 Such d 6 -complexes find applications as electroswitchable 20 emissive systems, 5a cellular imaging agents, 5b,c chromophores for photoredox chemistry, 5d etc. It would therefore be of great interest to develop chiral analogues 6 in order to benefit from the chiral version of emission, namely circularly polarized luminescence (CPL) which may potentially be used in cryptography or for 3D-displays. 7,[START_REF] Field | CPL active helicenes: a)[END_REF] In this communication, we describe the synthesis of tricarbonyl Re I complexes of general formula [Re(N^N')(CO) 3 X] 0/+ (X = halide, pyridyl or isocyanide) with N^N' being either achiral 3-(2pyridyl)-4-aza[4]-helicene (1a) or chiral 3-(2-pyridyl)-4-aza[6]helicene (M-and P-1b) (Scheme 1). The stereochemical features of these novel d 6 -complexes are presented in detail. The chiroptical properties of enantio-enriched samples and the nonpolarized and circularly polarized phosphorescence were measured experimentally and analyzed using quantum-chemical calculations.

Re I -complex 2a was obtained in 85% yield as a yellow-orange precipitate upon refluxing a solution of 1a 8g and Re(CO) 5 Cl in toluene for 5 hours (Scheme 1). It was fully characterized by multinuclear NMR spectroscopy (one set of peaks), by elemental analysis, UV-vis and emission spectroscopies. As compared to ligand 1a, the 1 H NMR spectrum of 2a shows strongly deshielded signals (except for H 2 , H 12 and H 6' ) with  up to +0.8 ppm for H 5 (see Supporting Information, SI). The UV-vis spectrum of ligand 1a in CH 2 Cl 2 displays a strong band at 295 nm ( > 50 x 45 10 3 M -1 cm -1 ), accompanied by several structured bands of lower intensity between 300 and 400 nm. Meanwhile, complex 2a shows several absorption bands between 230 and 370 nm (~ 30-43 x 10 3 M -1 cm -1 ) that can be assigned to intraligand π-π* transitions and a broad, low-energy absorption band between 370 and 480 nm (λ max = 398 nm, ε = 12700 M -1 cm -1 ) related to the incorporation of the Re I metal and predominantly assigned as ILCT with small contributions of MLCT character (vide infra).

The absorption maximum at 398 nm appears red-shifted compared to the corresponding band in Re(2,2'-bpy)(CO) 3 Cl 3f 55 (350 nm) indicating extended π-conjugation. Re I complex 2a is red-phosphorescent in CH 2 Cl 2 at RT ( max phos = 678 nm,  = 0.11%,  = 25 ns, see SI). The phosphorescence originates from the triplet charge-transfer state. It is facilitated by spin-orbit coupling at the rhenium heavy atom and 80 bathochromically shifted compared to that of Re(2,2'-bpy) (CO) 3 Cl ( max phos = 610 nm). 3f At 77 K, the phosphorescence of 2a is significantly shifted to shorter wavelengths ( max phos = 550 nm, = 7.9 s). Such a hypsochromic shift is usually explained by inversion in energies of 3 * and 3 MLCT triplet states and/or 85 by rigidification of the system. [START_REF] Ak | [END_REF] Note that as usual in this class of complexes, the quantum yield at RT is rather low. ii)

or iii) N N (±)-3a Re CO OC CO PF6 - N C + i) N N N N 1a (±)-2a 2 3 4 2' 6' Re CO Cl OC CO 1 12 11 N N (±)-4a Re CO OC CO BF4 - N + + N N (M,CRe)-2b 2 Re Cl CO OC CO ii) N N (M,ARe)-3b 1 Re OC CO PF6 - CO + + (M,CRe)-3b 2 N C
Re atoms adopt a slightly distorted octahedral geometry, with three carbonyl groups being fac-oriented around the Re I , as classically seen in such rhenium(I) tricarbonyl diimine complexes. [START_REF] Ak | [END_REF] The equatorial planes are defined by the chelate bipyridine ligand and two trans carbonyls. A third carbonyl and either the chlorine, the isocyanide or the pyridine are placed in the apical positions. Note that in structures 3a,4a the [4]helicenebpy ligand exhibits a helicity angle (defined as angle between the terminal rings of the helicene moiety) of ~35º and the cyanide and pyridine ligand are directed towards it, thus defining the (P,A Re ) and (M,C Re ) stereochemistry. [START_REF] Si | Stereochemistry of Coordination Compounds[END_REF] However, in solution, the helicene is not configurationally stable, and the Re center readily epimerizes (vide infra). As expected, the charged complexes displayed improved photophysical properties with similar UVvisible and emission spectra as for 2a (see SI), but with higher quantum yields (3a: 16%; 4a: 8.3%). These results prompted us to prepare tricarbonylrhenium(I) complexes bearing a configurationally stable enantiopure [6]helicene-bipy ligand.

Racemic 1b was reacted with Re(CO) 5 Cl in refluxing toluene for 5 hours, yielding after purification by column 20 chromatography two distinct diastereomeric Re(I) complexes (2b 1 and 2b 2 , with 28% and 52% yields, respectively) as evidenced by 1 H and 13 C NMR spectroscopy (for example H 15 : 6.7 ppm for 2b 1 and 6.9 ppm for 2b 2 , see SI). Complex 2b 2 crystallizes in a centrosymmetric space group (P2 1 /C) in which two enantiomeric structures, namely (M,C Re )-and (P,A Re )-2b 2 are present (Scheme 1). [START_REF] Si | Stereochemistry of Coordination Compounds[END_REF] Note that a substantial distortion results from the bite angles between the chelating N atoms of the helicenic ligand, the rhenium centre and the chloride ligand ranging between 82.6 and 84.3°. In complex 2b 2 the chlorine atom is directed towards the helicene moiety, whereas it directs outwards from the helicene core in the enantiomeric complexes (M,A Re ) and (P,C Re )-2b 1 . The helicity of the aza[6]helicene moiety ranges between 47.0-66.2°, which is typical for aza[6]helicene derivatives (58° for carbo[6]helicene). [START_REF] Field | CPL active helicenes: a)[END_REF] Finally, complexation with Re affords an extended -conjugation over the whole molecule, as evidenced by the small NCCN dihedral angles between the two chelating pyridine moieties (-3.1-6.0°). The extended -conjugation and the metal-ligand interaction are evidenced by UV-vis spectroscopy since 2b 1,2 display similar 40 UV-vis spectra with a set of several bands between 330 and 450 nm (~ 7-25  10 3 M -1 cm -1 ) that are bathochromically shifted and more intense compared to ligand 1b, together with a very weak band observed between 450 and 500 nm (see Figure S21). Calculations at the BHLYP/SV(P) level with the continuum solvent model for CH 2 Cl 2 reproduce well these data and show that the low-energy band of the spectrum is dominated by an ILCT transition, helicene) → *(N^N'), while the mediumenergy bands are mostly -to-* 'CT-like' transitions localized within the helicene moiety (vide infra, SI) in agreement with 50 assignments of absorption spectra of related rhenium(I) systems, in particular for complexes with large -conjugated ligands. 4c-e The overall contribution of the Re orbitals is low, meaning that the primary effect of the metal is to rigidify the system and induce strong charge-transfer from the helical -system to the 55 bipy N^N' part of the ligand. The simulated spectral shapes and band positions agree well with experiment. It is possible, though, that the overall involvement of Re orbitals in the absorption transitions is somewhat underestimated by the BHLYP functional (vide infra). Re I complexes 2b 1,2 are red-phosphorescent emitters 60 in CH 2 Cl 2 at RT (2b 1 :  max phos = 680 nm,  = 0.13%, = 27 ns; 2b 2 :  max phos = 673 nm,  = 0.16%, = 33 ns; for details see SI).

At 77 K, these complexes display phosphorescence at shorter wavelengths (2b 1 :  max phos = 560 nm, = 46 s; 2b 2 :  max phos = 554 nm, 43 s) (vide supra). Note that the emission properties 65 of diastereomers 2b 1,2 are only slightly different and (for  and ) within the uncertainty in the measurements (see SI).

Enantiopure complexes (M,A Re )-2b 1 and (M,C Re )-2b 2 were then prepared from enantiopure M-1b (their mirror-images (P,C Re )-2b 1 and (P,A Re )-2b 2 from P-1b). Enantiopure complexes 2b 1,2 2b 1 shows a strong negative band around 261 nm ( = -114 M -1 cm -1 ) and strong positive ones at 350 (+81 M -1 cm -1 ) and 368 nm (+76 M -1 cm -1 ) accompanied by weaker bands between 380 and 450 nm (20-40 M -1 cm -1 ) and an even weaker one around 480 nm but of opposite sign ( ~ -0.6 M -1 cm -1 ). Diastereomeric 80 complex (P,A Re )-(+)-2b 2 exhibits the same ECD active bands as 2b 1 but they are more intense. A comparison with experimental ECD of 1a enantiomers is displayed in Figure 1. The calculated (BHLYP/SV(P) with the continuum solvent model for CH 2 Cl 2 ) spectra of 2b 1,2 qualitatively agree well with the experimental 85 results (Figure 3 and S5). A detailed analysis of dominant excitations in the low-and medium-energy parts of the simulated spectra of 2b 1,2 indicates that the low-energy tail of the first positive ECD band is caused by excitation no. 1 calculated at E = 3.3 eV (375 nm). The excitation can be assigned as a -* ILCT 90 transition involving the helicene-centered HOMO (H), H-1, and the bipyridine N^N'-centered LUMO (L), for example for 2b 1 : H-L 51% and H-1-L 18% (see Figure 3 and SI). The second dominant 2b 1,2 excitation is no. 5 calculated at E = 3.8 eV (330 nm) with the strongest rotatory strength. It involves two main 95 contributions from  and * orbital pairs localized mostly in the helicene moiety: H-L+1 and H-1-L+1 (respectively 43% and 25% for 2b 1 ). The excitation reveals partial CT character.

A novel aspect of these rhenium(I) helicene-based complexes is that they are CPL active (Figure 2, top panels). 7,[START_REF] Field | CPL active helicenes: a)[END_REF] To the best of 100 our knowledge, these are the first examples of CPL-active phosphorescent rhenium complexes. Indeed phosphorescent (P,A Re ) and (M,C Re )-2b 2 enantiomers displayed mirror-imaged CPL spectra (Figure 2) with opposite g lum values ((P,A Re )-2b 2 : +3.1  10 -3 and (M,C Re )-2b 2 : -2.8  10 -3 ) around the emission 105 maximum (~670 nm). These values are of the same order as for the 1b ligand enantiomers (g lum ~±10 -3 ) but lower than those of previously published platina[6]helicenes (g lum ~ ±10 -2 ), 8e because Re orbitals are less involved in the helical -system of the molecule (vide supra). In order to improve the efficiency of the chiroptical and photophysical properties, tricarbonyl-isocyanide-helicene-bipy-Re I complex M-3b was prepared (see Scheme 1) in 75% yield from either (M,A Re )-(-)-2b 1 or (M,C Re )-(-)-2b 2 . In this complex, 20 the Re center appeared labile and 3b was obtained as a mixture of (M,A Re )-3b 1 and (M,C Re )-3b 2 as observed by 1 H and 13 C NMR spectroscopy (diastereomeric ratio 50:50, see Figure S27) regardless of the diastereomeric purity of the starting compound used (either 2b 1 or 2b 2 or 2b 1,2 ). Nevertheless, as expected, this diastereomeric mixture displayed an improved quantum yield ( max phos = 598 nm,  = 6%, = 79 s; see SI) as compared to 2b 1,2 . The UV-vis spectrum of 3b 1,2 displays the same shape as 2b 1 (see Figure S21). Compared to (P,C Re )-2b 1 and (P,A Re )-2b 2 , cationic diastereomeric mixture of Re I complexes P-3b 1,2 displays an additional positive CD-active band around 450 nm ( = 17.5 M -1 cm -1 ). As for 2b 1,2 , this latter band does not involve the Re center, but corresponds to the H-L transition (>74%) with strong charge transfer from the -helicene to the bipy moiety, as evidenced by BHLYP calculations (see Figure 3 35 and SI). The appearance of the 450 nm band is caused mainly by a bathochromic shift of the first singlet excitation. This charge transfer excitation is likely responsible for the enhancement of molar rotations as compared to 2b 1,2 {(P,AC Re )-(+)-3b:   23

D  = 15040 (  5%) (C = 8.8  10 -5 M, CH 2 Cl 2 ); (M,AC Re )-(-)-3b:   23 D  40 = -14230 (  5%) (C = 9.7  10 -5 M
, CH 2 Cl 2 ); calc. BHLYP Boltzmann average for 3b 1,2 conformers is +14034 degree cm 2 /dmol for the P-isomers, see SI}.

Quantum-chemical calculations of luminescence properties have been performed for 2b 1,2 and 3b 1,2 . The results support the experimental assignments: The energies of T 1 → S 0 phosphorescence transitions (~2.1 eV) are similar for both 2b 1,2 and 3b 1,2 and agree fairly well with the experimental data (Table S5). An overestimation of the calculated versus measured energies is consistent with a blue-shift of calculated 2b 1,2 and 3b 1,2 absorption and CD spectra. The emission energies from spin-orbit (SO) calculations agree with non-SO calculations but the former allow predictions of the phosphorescence lifetimes. Application of the BHLYP functional along with the Tamm-Dancoff approximation (see SI) resulted in much too high emission lifetimes (Table S6). As the involvement of Re orbitals facilitates the formally spin-forbidden T 1 → S 0 phosphorescence transitions via spin-orbit coupling, decreasing the corresponding lifetimes, too high  calculated with BHLYP may indicate that the metal orbital contributions to the frontier MOs are somewhat too 60 small. The performance of a given functional for singlet vs. triplet transitions is not necessarily the same. When applying a computational protocol for emission lifetimes devised recently by Mori et al. 11 for organometallic complexes (full TDDFT with the B3LYP functional), a dramatic improvement of the lifetimes and 65 some lowering of the emission energies (to ~1.9 eV) was obtained (Table S7), which correlates with increased participation of Re orbitals in the frontier MOs at the triplet geometries. Notably the experimental trend of an increase in emission lifetime by roughly an order of magnitude when going from 2b 1,2 70 to 3b 1,2 is correctly reproduced with B3LYP and qualitatively consistent with lesser metal orbital participation (lesser MLCT character) in the T 1 emission transitions for 3b 1,2 as compared to 2b 1,2 (see SI). $,4a,b Finally, mirror-imaged CPL spectra were obtained in CH 2 Cl 2 for (M,AC Re )-3b 1,2 and (P,AC Re )-3b 1,2 (Figure 2) with respective g lum values of -0.0015 and +0.0013. Overall, cationic Re I 95 complexes display similar CPL characteristics as neutral ones, but combined with a higher quantum yield, the polarized emitted light is stronger. Although the Re d orbitals are not strongly involved in the electronic  systems of these novel metallohelicenes, the metal helps to increase the-conjugation pathway 100 and promotes charge transfer excitations within the -helical ligand. In addition, the presence of the rhenium heavy atom makes these complexes chiral phosphors with unprecedented CPL activity.

In conclusion, the first CPL-active rhenium complexes have

  Saleh, a Monika Srebro, b Thibault Reynaldo, a Nicolas Vanthuyne, c Loïc Toupet, a Victoria Y. Chang, d G. Muller, d J. A. Gareth Williams, e Christian Roussel, c Jochen Autschbach,* ,f Jeanne Crassous* ,a

60 65 Scheme 1 .

 651 Scheme 1. Synthesis of neutral and charged rhenium complexes 2a-4a 70

  70 display similar molar rotation (MR) values to 1b in CH 2 Cl 2 {(P,C Re )-(+)-2b 1 :   23 D  = +9260 degree cm 2 /dmol (±5%), calc. BHLYP +12721; (P,A Re )-(+)-2b 2 :   23 D  = +10260 (±5%), calc. BHLYP +11888 degree cm 2 /dmol; P-(+)-1b: 8g   23 D  = +12000 (±5%), calc. BHLYP +14176, see SI}. The ECD spectrum of P-75

  110

Figure 1 .

 1 Figure 1. Experimental CD spectra of enantiopure M-(-)-(dotted red) and P-(+)-1b (plain red) and their corresponding enantiopure Re I complexes (M,A Re )-(-)-2b 1 and (P,C Re )-(+)-2b 1 (light blue) and (M,C Re )-(-)-2b 2 and (P,A Re )-(+)-2b 2 (dark blue). Inset: CD spectra of (-)-and (+)-2b 1 between 450-550 nm.

Figure 2 .

 2 Figure 2. CPL (upper curves within each panel) and total luminescence (lower curves within each panel) spectra of M-(-)-1b (top left red), P-(+)-1b (top left black), M-(-)-2b 2 (top right red), P-(+)-2b 2 (top right black), M-(-)-3b 1,2 (bottom left red), P-(+)-3b 1,2 (bottom left black) in degassed 90

  105been prepared by coordination of enantiopure[6]helicenebipyridine type ligands. Their chiroptical and photophysical properties (optical rotation, electronic circular dichroism, nonpolarized and circularly polarized luminescence, quantum yields, phosphorescence lifetimes) have been studied and analyzed with 110 the help of quantum-chemical calculations. 1265833 to J.A.) and the Foundation for Polish Science Homing Plus programme co-financed by the European Regional Development Fund (to M.S.). M.S. is also grateful for financial support via a Ministry of Science and Higher Education in Poland scholarship. We acknowledge the Center for Computational Research (CCR) at the University at Buffalo for their continuing support of our research projects. G.M. thanks the National Institute of Health, Minority Biomedical Research Support (1 SC3 GM089589-05 and 3 S06 GM008192-27S1) and the Henry Dreyfus Teacher-Scholar Award for financial support.

Figure 3 .

 3 Figure 3. Left: experimental CD spectra of enantiopure complexes (P,C Re )-(+)-2b 1 (light blue), (P,A Re )-(+)-2b 2 (dark blue) and of (M, CA Re )-(-)-3b 1,2 (dotted green) and (P, CA Re )-(+)-3b 1,2 (plain green). Right: calculated CD spectra of (P,C Re )-2b 1 , and (P,A Re )-2b 2 and Boltzmann-averaged spectrum for P-3b 1,2 conformers. No spectral shift has been applied. View of HOMO and LUMO of 2b 1 and 3b 1 (0.04 au). First excitation energies indicated by dots on the abscissa.Notes and referencesa Institut des Sciences Chimiques de Rennes, UMR 6226, Institut de Physique de Rennes, UMR 6251, Campus de Beaulieu, CNRS-Université de Rennes 1, 35042 Rennes Cedex, France. E-mail:

Acknowledgments

We thank the Ministère de l'Education Nationale, de la

Recherche et de la Technologie, the Centre National de la Recherche Scientifique (CNRS), the ANR (10-BLAN-724-1-NCPCHEM and 12-BS07-0004-METALHEL-01) and the LIA Rennes-Durham for financial support. This work has received financial support from the National Science Foundation (CHE # Note that enantiopure (+) and (-) complexes (1a)Re(CO)3Br were prepared by chiral HPLC and they displayed very weak CD activity ( max = 20 M -1 cm -1 , see Figure S23). $ The direct comparison between experimental and calculated lifetimes must be treated with some caution, as the former are also affected by non-50 radiative decay pathways, which may be non-negligible even at 77 K. We assume that the non-radiative decay rates for the complexes are similar under these conditions.


avec admiration. Marc Yor
email: philippe.bougerol@upmc.fr

THE MATSUMOTO AND YOR PROCESS AND INFINITE DIMENSIONAL HYPERBOLIC SPACE

come

The Matsumoto and Yor process and infinite dimensional hyperbolic space

1. Introduction 2. The Matsumoto -Yor process 3. Brownian motion on hyperbolic spaces 3.1. Hyperboloid model 3.2. Upper half space model 3.3. Ground state processes 3.4. Matsumoto -Yor process as a limit 3.5. A conditional law 4. Infinite series of symmetric spaces 4.1. SU (p, q) 4.2. Cartan decomposition and radial part 4.3. Iwasawa decomposition and horocyclic coordinates 4.4. Brownian motion on G/K and infinite Brownian loop 4.5. Distinguished Brownian motion on S 4.6. Limit as q → +∞ 4.7. SO(p, q) and Sp(p, q) 5. Generator of (η t ) 5.1. Inozemtsev limit to Quantum Toda Hamiltonian 5.2. Asymptotics for SU (p, q) 6. Series of homogeneous trees 7. Appendix : Asymptotics of spherical functions in rank one References

Introduction

The aim of this paper is mainly to see in the first part that the Matsumoto -Yor process [START_REF] Matsumoto | A version of Pitman's 2M -X theorem for geometric Brownian motions[END_REF] η t = t 0 e 2Bs-Bt ds, t ≥ 0, where (B t ) is a standard Brownian motion, appears naturally as the radial part of the Brownian motion on the hyperbolic space H q at the bottom of the spectrum when the dimension q → +∞. Marc Yor [40] asked in 1999 whether there is a geometric interpretation of this process which provides a direct proof of its Markovianity. Notice that t 0 e µBs-Bt ds, t ≥ 0, is a Markov process only for µ = 1 and µ = 2 (this follows from [START_REF] Matsumoto | Markov or non-Markov property of cM -X processes[END_REF] by a scaling and limit argument). The case µ = 1 is easy since it follows from Ito's formula that this process is solution of a stochastic differential equation. Things are more subtle for µ = 2 because the process is not Markov for the filtration σ(B s , 0 ≤ s ≤ t).

In a second part, we describe the limit of the radial part of the Brownian motion for the three infinite series of symmetric spaces of higher rank, namely SO(p, q), SU (p, q), Sp(p, q) when p is fixed and q → +∞. For SU (p, q) we give the generator of the limit by studying the asymptotic behaviour of spherical functions (such an analysis is not available yet for the other cases).

In the higher rank case, these processes are different from the Whittaker processes obtained by O'Connell [START_REF] O'connell | Directed polymers and the quantum Toda lattice[END_REF] and Chhaibi [START_REF] Chhaibi | Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion[END_REF] who also generalize the Matsumoto -Yor process for real split semi simple groups, but in a different (and more interesting) direction linked with representation theory and geometric crystals.

In the last part we show that in the q-adic case of rank one, or more generally on regular trees T q , the radial part of the simple random walk at the bottom of the spectrum converges when q → ∞ to the discrete Pitman process 2 max

0≤k≤n Σ k -Σ n , n ∈ N,
where (Σ n ) is the simple random walk on Z. Contrary to the real case we use here only elementary arguments and the treatment is self-contained.

In an appendix we describe how to modify Shimeno [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF] in order to obtain the needed asymptotics for spherical functions in rank one.

The Matsumoto -Yor process

Let us first recall a now classical theorem of Pitman [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF]. Let B t , t ≥ 0, be a standard real Brownian motion starting at 0. Theorem 2.1 (Pitman, [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF]). The process 2 max 0≤s≤t B s -B t , t ≥ 0, is a Markov process on R + . It is the Bessel(3) process with generator 1 2

d 2 dr 2 + 1 r d dr .
In 1999, Matsumoto and Yor [START_REF] Matsumoto | A version of Pitman's 2M -X theorem for geometric Brownian motions[END_REF][START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener Functionals, Part I: A time-inversion approach[END_REF][START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals, Part II: The role of the generalized inverse Gaussian laws[END_REF], have found the following exponential generalization of Pitman's theorem.

Theorem 2.2 (Matsumoto and Yor, [START_REF] Matsumoto | A version of Pitman's 2M -X theorem for geometric Brownian motions[END_REF]). The process η t = t 0 e 2Bs-Bt ds, t ≥ 0, is a Markov process and the generator of log η t is 1 2

d 2 dr 2 + ( d dr log K 0 (e -r )) d dr ,
where K 0 is the Macdonald function.

Recall that, for λ ∈ R, the classical Macdonald function K λ is

K λ (x) = 1 2 ( x 2 ) λ
∞ 0 e -t e -x 2 /4t t 1+λ dt.

A geometric intuition of the Markovianity of this process does not follow clearly from the rather intricate known proofs (either the original ones ( [START_REF] Matsumoto | A version of Pitman's 2M -X theorem for geometric Brownian motions[END_REF][START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener Functionals, Part I: A time-inversion approach[END_REF]) or Baudoin [START_REF] Baudoin | Further exponential generalization of Pitman's 2M -X theorem[END_REF], see also [START_REF] Baudoin | Exponential functionals of Brownian motion and classone Whittaker functions[END_REF]). Pitman's theorem can be recovered by Brownian scaling and Laplace's method.

Brownian motion on hyperbolic spaces

3.1. Hyperboloid model. There are many realizations of the hyperbolic spaces (see for instance Cannon et al. [START_REF] Cannon | Hyperbolic Geometry[END_REF]). We will consider two of them: the hyperboloid and the upper half-space models. A very convenient reference for us is Franchi and Le Jan's book [START_REF] Franchi | Hyperbolic Dynamics and Brownian Motion, An Introduction[END_REF], which we will follow. A more Lie theoretic approach will be applied in Section 4 for the higher rank case (and of course could also be used here). For ξ, ξ

′ ∈ R q+1 let ξ, ξ ′ = ξ 0 ξ ′ 0 - q k=1 ξ k ξ ′ k .
The hyperbolic space H q of dimension q is

H q = {ξ ∈ R q+1 ; ξ, ξ = 1, ξ 0 > 0},
which is the upper sheet of an hyperboloid, with the Riemannian metric d defined by cosh d(ξ, ξ ′ ) = ξ, ξ ′ .

The group SO 0 (1, q) is the connected component of the identity in

{g ∈ Gl(q + 1, R); gξ, gξ ′ = ξ, ξ ′ , ξ, ξ ′ ∈ R q+1 }.
It acts by isometry on H q by matrix multiplication on R q+1 . Let {e 0 , • • • , e q } be the canonical basis of R q+1 . We consider e 0 as an origin in H q and we let o = e 0 . The subgroup

K = {g ∈ SO o (1, q); ge 0 = e 0 } is isomorphic to SO(q), it is a maximal compact subgroup of SO o (1, q) and
H q = SO o (1, q)/K.
Each ξ ∈ H q can be written uniquely as

ξ = (cosh r)e 0 + (sinh r)ϕ,
where r ≥ 0, ϕ ∈ S q-1 = { q k=1 ϕ k e k ; q k=1 ϕ 2 k = 1}. One has d(o, ξ) = r and (r, ϕ) are called the polar coordinates of ξ. The Laplace Beltrami operator ∆ on H q is given in these coordinates by (see [START_REF] Franchi | Hyperbolic Dynamics and Brownian Motion, An Introduction[END_REF]Prop. 3.5.4]

) (3.1) ∆ = ∂ 2 ∂r 2 + (q -1) coth r ∂ ∂r + 1 sinh 2 r ∆ ϕ S q-1 ,
where ∆ ϕ S q-1 is the Laplace operator on the sphere S q-1 acting on the ϕvariable.

We denote (ξ t ) the Riemannian Brownian motion on H q , defined as the diffusion process with generator ∆/2, starting from the origin o = e 0 . Since the group K acts transitively on the spheres {ξ ∈ H q ; d(o, ξ) = r}, d(o, ξ t ) is a Markov process, called the radial part of ξ t . It follows from (3.1) that its generator is ∆ R /2 where

(3.2) ∆ R = d 2 dr 2 + (q -1) coth r d dr .
3.2. Upper half space model. We introduce the upper half space model following Franchi and Le Jan [START_REF] Franchi | Hyperbolic Dynamics and Brownian Motion, An Introduction[END_REF]: we consider the square matrices Ẽj , 1 ≤ j ≤ 1 + q, of order q + 1 given by the expression

t Ẽ1 + q-1 j=1 x j Ẽj+1 =         0 t x 1 x 2 .. x q-1 t 0 x 1 x 2 .. x q-1 x 1 -x 1 0 0 .. 0 x 2 -x 2 0 0 .. 0 .. .. .. .. .. .. x q-1 -x q-1 0 0 .. 0         , when t, x 1 , • • • , x q-1 ∈ R. Then S = {t Ẽ1 + q-1 j=1 x j Ẽj+1 , t ∈ R, x ∈ R q-1 }
is a solvable subalgebra of the Lie algebra of SO 0 (1, q). For x ∈ R q-1 and y > 0 let

T x,y = exp( q-1 j=1
x j Ẽj+1 ) exp((log y) Ẽ1 ). [START_REF] Franchi | Hyperbolic Dynamics and Brownian Motion, An Introduction[END_REF]Proposition 1.4.3]). Therefore the map (x, y) → T x,y is an isomorphism between the affine group of R q-1 , namely the semi-direct product R q-1 ⋊ R * + , and the group S. One has ([15, Proposition 2.1.3, Corollary 3.5.3]), Proposition 3.1. The map T : R q-1 × R * + → H q given by T (x, y) = T x,y e 0 is a diffeomorphism. In these so-called horocyclic or Poincaré coordinates (x, y) ∈ R q-1 × R * + , the hyperbolic distance is given by

Then S = {T x,y ; (x, y) ∈ R q-1 × R * + } is the Lie subgroup of SO o (1, q) with Lie algebra S. Moreover, T x,y T x ′ ,y ′ = T x+yx ′ ,yy ′ (see
cosh d( T (x, y), T (x ′ , y ′ )) = x -x ′ 2 + y 2 + y ′2 2yy ′ .
The pull back of the Laplace Beltrami operator ∆ on H q is

y 2 ∂ 2 ∂y 2 + (2 -q)y ∂ ∂y + ∆ x q-1 ,
where ∆ x q-1 is the Euclidean Laplacian of R q-1 acting on the coordinate x. In horocyclic coordinates on H q , the hyperbolic Brownian motion ξ t has a nice probabilistic representation which comes from the fact that it can be seen as a Brownian motion on the group S where Definition 3.2. On a Lie group a process is called a Brownian motion if it is a continuous process with independent stationary multiplicative increments.

Indeed, let (X t , Y t ) ∈ R q-1 × R *
+ be the horocyclic coordinates of ξ t , and let ̺ = q -1 2 , then (see [START_REF] Bougerol | Exemples de théorèmes locaux sur les groupes résolubles[END_REF], or [15, Theorem 7.6.5.1]):

Proposition 3.3. One can write X t = t 0 e B -ρ s dW (q-1) s , Y t = e B -ρ t ,
where

B -ρ t = B t -t̺ is a real Brownian motion on R with drift -̺ and W (q-1) s is a q -1-dimensional standard Brownian motion, independent of B -̺ .
Proof. This follows immediately from Ito's formula and the expression of the generator given in Proposition 3.1.

3.3.

Ground state processes.

3.3.1.

Ground state processes on a manifold. We will need the notion of ground state process. In order to introduce it rapidly we use the set up presented in Pinsky [START_REF] Pinsky | Positive harmonic functions and diffusion[END_REF] (see also Pinchover [START_REF] Pinchover | Large time behaviour of the heat kernel and the behaviour of the Green function near criticality for nonsymmetric elliptic operators[END_REF]). On a manifold M we consider an elliptic operator which can be written locally as

D = i,j a ij (x) ∂ 2 ∂x i ∂x j + i b i (x) ∂ ∂x i + V (x),
where the coefficients are C ∞ and the matrix a is symmetric positive definite (i.e. hypothesis H loc in [34, p. 124]). Let

C D = {u ∈ C ∞ (M ); Du(x) = 0, u(x) > 0, for all x ∈ M }.
There exists λ 0 ∈ (-∞, +∞] such that, for any λ < λ 0 , C D-λ is empty, and for λ > λ 0 , C D-λ is not empty ([34, Theorem 4.3.2]). For a self adjoint operator -λ 0 coincide with the bottom of the spectrum on L 2 under general conditions ([34, Proposition 4.10.1]), for instance for the Brownian motion on a Riemannian manifold.

Definition 3.4. When λ 0 < +∞, λ 0 is the generalized principal eigenvalue of D.

A positive function h ∈ C ∞ (M ) such that Dh = λ 0 h is called a ground state (it does not always exist and is in general not unique). The Doob h-transform of D is the operator D h defined by

D h f = 1 h D(hf ) -λ 0 f.
The associated Markov process is called the h-ground state process. When a is the identity matrix, then

(3.3) D h = D + 2 i ∂ log(h) ∂x i ∂ ∂x i ,
(see [START_REF] Pinsky | Positive harmonic functions and diffusion[END_REF]Section 4.1]). For the Brownian motion on a Riemannian manifold, when V = 0 and once an origin is fixed, there is sometimes a canonical choice of a ground state for which the ground state process has a probabilistic interpretation as an infinite Brownian loop. This infinite Brownian loop is, loosely speaking, the limit of the first half of the Brownian bridge around the origin when its length goes to infinity (see [START_REF] Anker | The infinite Brownian loop on a symmetric space[END_REF] for details).

3.3.2.

A radial ground state process on H q . In Proposition 3.3 the component Y t of the Brownian motion on H q depends on the dimension q only through the drift ̺ = (q-1)/2. In order to see what happens when q → +∞, it is natural to first kill this drift. This is why we will consider the Brownian motion at the bottom of its spectrum.

The generalized principal eigenvalue of the Laplace Beltrami operator ∆ is -̺ 2 and there exists a unique radial function ϕ 0 on H q , called the basic Harish Chandra function, such that ∆ϕ 0 = -̺ 2 ϕ 0 and ϕ 0 (o) = 1 (see, e.g., Davies [13,5.7.1], Gangolli and Varadarajan, [START_REF] Gangolli | Harmonic Analysis of Spherical Functions on Real Reductive Groups[END_REF], Helgason [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF]).

Definition 3.5. The infinite Brownian loop on H q is the ϕ 0 -ground state process (ξ 0 t ) with generator

1 2 ∆ ϕ 0 f = 1 2ϕ 0 ∆(ϕ 0 f ) + ̺ 2 2 f.
The function ϕ 0 is radial, which means that ϕ 0 (ξ) is a function of r = d(o, ξ); we define φ0 : R → R by, φ0 (r) = ϕ 0 (ξ).

It follows from (3.2) and (3.3) that, Proposition 3.6. The process {d(o, ξ 0 t ), t ≥ 0} is a Markov process on R + with generator ∆ φ0 R /2 where

∆ φ0 R = d 2 dr 2 + ((q -1) coth r + 2 φ′ 0 (r) φ0 (r) ) d dr .

3.3.3.

A non radial ground state process on H q . Although our main interest is in ξ 0 t we will actually need another ground state process for which computations are easier. It is not invariant under rotations but its radial part is the same as the one of ξ 0 t (see Proposition 3.8 below). Let Ψ 0 : H q → R be the function defined by, if (x, y) ∈ R q-1 × R * + are the horocyclic coordinates of ξ ∈ H q , (3.4) Ψ 0 (ξ) = y (q-1)/2 = e ̺ log y .

Notice that Ψ 0 (o) = 1 and

∆Ψ 0 (ξ) = y 2 ∂ 2 ∂y 2 y ̺ -(q -2)y ∂ ∂y y ̺ = -̺ 2 Ψ 0 (ξ).
Thus Ψ 0 is, like ϕ 0 , a positive ground state of ∆. Let {S t , t ≥ 0} be the Ψ 0 -ground state process of the Brownian motion (ξ t ) on H q . By definition, for all T > 0, when f :

C([0, T ], H q ) → R + is measurable, (3.5) E(f (S t , 0 ≤ t ≤ T )) = e ̺ 2 T /2 E(f (ξ t , 0 ≤ t ≤ T ) Ψ 0 (ξ T ) Ψ 0 (ξ 0 ) )
where Ψ 0 (ξ 0 ) = 1 since ξ 0 = o. Then, it is easy to see that: [START_REF] Bougerol | Exemples de théorèmes locaux sur les groupes résolubles[END_REF]). In horospherical coordinates,

Lemma 3.7 ([
S t = ( t 0 e Bs dW (q-1) s , e Bt ),
where B t is a standard (i.e. driftless) one dimensional Brownian motion, and W (q-1) is a standard q -1 dimensional Brownian motion, independent of B.

Let dk be the Haar measure on K, normalized as a probability measure. The function

K Ψ 0 (k • ξ) dk
is a positive radial eigenvector of ∆ with eigenvalue -̺ 2 . Therefore, by uniqueness, we have the well known formula of Harish Chandra (see Helgason, [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF])

ϕ 0 (ξ) = K Ψ 0 (k • ξ) dk.
The processes S t and ξ 0 t do not have the same law, and S t is not rotation invariant. However, Proposition 3.8. The two processes {d(o, S t ), t ≥ 0} and {d(o, ξ 0 t ), t ≥ 0} have the same law.

Proof. By invariance under rotation of the Brownian motion on H q , for any k ∈ K, the processes (ξ t ) and (k•ξ t ) have the same law. Hence, after integration over K, for T > 0 and any measurable function

f : C([0, T ], R + ) → R + , E(f (d(o, S t ),0 ≤ t ≤ T )) = e ̺ 2 T /2 E(f (d(o, ξ t ), 0 ≤ t ≤ T )Ψ 0 (ξ T )) = e ̺ 2 T /2 E(f (d(o, k • ξ t ), 0 ≤ t ≤ T )Ψ 0 (k • ξ T )) = e ̺ 2 T /2 E(f (d(o, ξ t ), 0 ≤ t ≤ T )Ψ 0 (k • ξ T )) = e ̺ 2 T /2 E(f (d(o, ξ t ), 0 ≤ t ≤ T ) K Ψ 0 (k • ξ T ) dk) = e ̺ 2 T /2 E(f (d(o, ξ t ), 0 ≤ t ≤ T )ϕ 0 (ξ T )) = E(f (d(o, ξ 0 t ), 0 ≤ t ≤ T )).
Remark 3.9. In horocyclic coordinates in (3.4) the function (x, y) → log y is the Busemann function on H q associated with the point at infinity y = +∞ and Ψ 0 is a minimal eigenfunction. The process (S t ) can be interpreted as the Brownian motion on H q conditioned to have 0 speed (i.e. d(o, S t )/t → 0 as t → +∞) and to exit at y = +∞ (see [START_REF] Guivarc | Compactifications of symmetric spaces[END_REF]).

3.4. Matsumoto -Yor process as a limit. Our main result is the following.

Theorem 3.10. As q → +∞, the process

d(0, ξ 0 t ) -log q, t > 0,
converges in distribution to log η t , t > 0, where

η t = t 0 e 2Bs-Bt ds
is the Matsumoto -Yor process, which is therefore a Markov process.

Proof. By Proposition 3.8, it is enough to show that, almost surely,

(3.6) lim q→∞ d(o, S t ) -log q = log t 0 e 2Bs-Bt ds.
The origin o in H q is e 0 = T (0, 1). By Proposition 3.1,

cosh d(o, T (x, y)) = x 2 + y 2 + 1 2y , and cosh d(0, S t ) = t 0 e Bs dW (q-1) s 2 + e 2Bt + 1 2e Bt , thus 2 cosh d(0, S t ) q = e Bt + e -Bt q + e -Bt 1 q q-1 k=1 ( t 0 e Bs dβ (k) s ) 2 ,
where β

s , k ≥ 1, are independent standard Brownian motions. Conditionally on the σ-algebra σ(B r , r ≥ 0), the random variables Therefore, by the law of large numbers, a.s.

lim q→+∞ 1 q e d(0,St) = lim q→+∞ 2 q cosh d(0, S t ) = e -Bt t 0 e 2Bs ds.
The limit is Markov as a limit of Markov processes.

Let us recover the generator of the Matsumoto -Yor process. Let δ(r) = sinh q-1 r, then, by Proposition 3.6,

∆ φ0 R = d 2 dr 2 + 2 d dr log(δ 1/2 φ0 )(r) d dr , hence the generator of d(o, ξ 0 t ) -log q is (3.7) 1 2
d 2 dr 2 + d dr log(δ 1/2 φ0 )(r + log q) d dr .
Therefore the next proposition follows from Corollary 7.3 (since there, m α = q -1 for SO(1, q)).

Proposition 3.11 ([26]

). The generator of the Matsumoto -Yor process is 1 2

d 2 dr 2 + ( d dr log K 0 (e -r )) d dr .
Remark 3.12. We will clarify in Section 4 the occurrence of the ground state K 0 of the Toda operator d 2 dr 2 -e -2r . 3.5. A conditional law. The intertwining property which occurs in the proof of Matsumoto -Yor theorem can also be establish by our approach. Proposition 3.13 ([26]). When η t = e -Bt t 0 e 2Bs ds, and λ ∈ R,

E(e λBt |σ(η s , 0 ≤ s ≤ t)) = K λ (1/η t ) K 0 (1/η t ) .
Proof. For ξ ∈ H q with horocyclic coordinates (x, y) ∈ R q-1 ×R * + and λ ∈ R, let λ(ξ) = λ log y. We consider the Harish Chandra function ϕ λ (see [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF]Theorem IV.4.3]) given by

ϕ λ (ξ) = K e (λ+ρ)(k•ξ) dk,
(it of course depends on q) and we write φλ (r) = ϕ λ (ξ) when r = d(o, ξ). Since λ(S t ) = λB t and e ̺(ξ) = Ψ 0 (ξ), it follows from (3.5) that, when

f : C([0, t]) → R + is measurable, e -̺ 2 t/2 E(f (d(o, S s ), 0 ≤ s ≤ t))e λBt ) = e -̺ 2 t/2 E(f (d(o, S s ), 0 ≤ s ≤ t))e λ(St) ) = E(f (d(o, ξ s ), 0 ≤ s ≤ t)e (λ+ρ)(ξt) ) = E(f (d(o, ξ s ), 0 ≤ s ≤ t) K e (λ+ρ)(k•ξt) dk) = E(f (d(o, ξ s ), 0 ≤ s ≤ t)ϕ λ (ξ t )) = E(f (d(o, ξ s ), 0 ≤ s ≤ t) ϕ λ (ξ t ) ϕ 0 (ξ t ) ϕ 0 (ξ t )) = E(f (d(o, ξ s ), 0 ≤ s ≤ t) ϕ λ (ξ t ) ϕ 0 (ξ t ) e ρ(ξt) ) = e -̺ 2 t/2 E(f (d(o, S s ), 0 ≤ s ≤ t) φλ (d(o, S t )) φ0 (d(o, S t ))
).

We have again used the fact that for any k ∈ K, {ξ t , t ≥ 0} has the same law as {k • ξ t , t ≥ 0}. For each q,

E(f (d(0, S s )-log q, 0 ≤ s ≤ t))e λBt ) = E(f (d(0, S s ) -log q, 0 ≤ s ≤ t) φλ ((d(o, S t ) -log q) + log q) φ0 ((d(o, S t ) -log q) + log q) ).
We choose for f a continuous bounded function with bounded support. Letting q → +∞, it then follows from Theorem 3.10 and Corollary 7.3 that

E(f (log η s , 0 ≤ s ≤ t)e λBt ) = E(f (log η s , 0 ≤ s ≤ t) K λ (1/η t ) K 0 (1/η t ) ).
Remark

Infinite series of symmetric spaces

We will now consider the same problem as above for the infinite series of symmetric spaces of non positive curvature, when the rank is fixed and the dimension goes to infinity. We will see that in the rank one case one finds the same result as for hyperbolic spaces, but that new phenomenons occur in higher rank.

There are only three infinite series of (irreducible) Riemannian symmetric spaces of non positive (non zero) curvature namely the spaces G/K where G = SO(p, q), SU (p, q) and Sp(p, q), and K is a maximal compact subgroup (see [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF]). We will suppose that p ≤ q, then p is the rank of the symmetric space. 4.1. SU (p, q). We first consider the symmetric spaces associated with the series SU (p, q) with the rank p fixed, when q → +∞. As in the case of hyperbolic spaces, it is convenient to use two descriptions of SU (p, q). The usual description is the following (see, e.g., [START_REF] Hoogenboom | Spherical functions and invariant differential operators on complex Grassmann manifolds[END_REF], [START_REF] Knapp | Lie groups, beyond an introduction[END_REF]): SU (p, q) is the set of (p + q) × (p + q) matrices with entries in C and with determinant 1, which conserve the quadratic form

p i=1 z i zi - q j=1 z p+j zp+j .
Let I p and I q be the identity matrix of order p and q, and let J =

I p 0 0 -I q then SU (p, q) = {M ∈ Sl(p + q, C); M * JM = J}.
If we write M by block as

M = M 1 M 2 M 3 M 4 ,
where the size of the matrix

M 1 is p × p, M 2 is p × q, M 3 is q × p and M 4 is q × q, then M ∈ SU (p, q) when M * 1 M 1 -M * 2 M 2 = I p , M * 4 M 4 -M * 3 M 3 = I q , M * 1 M 3 = M * 2 M 4 , det(M ) = 1. 4.2.
Cartan decomposition and radial part. Let K be the following maximal compact subgroup of SU (p, q) (we put the notation K aside for a later use)

K = { K 1 0 0 K 4 ∈ Sl(p + q, C), K 1 ∈ U (p), K 4 ∈ U (q)}. When r = (r 1 , • • • , r p ) ∈ R p , let D p (r) =   cosh r sinh r 0 sinh r cosh r 0 0 0 I q-p   , where cosh r =     cosh r 1 0 • • • 0 0 cosh r 2 • • • 0 0 • • • • • • 0 0 0 • • • cosh r p     ,
and sinh r is the same matrix with cosh replaced by sinh. We consider the closed Weyl chamber

A + = {(r 1 , • • • , r p ) ∈ R p , r 1 ≥ r 2 ≥ • • • ≥ r p ≥ 0}.
The Cartan decomposition says that any M in SU (p, q) can be written as

M = k 1 D p (r)k 2 with k 1 , k 2 ∈ K and r ∈ A + . Such a
4.2. If M = M 1 M 2 M 3 M 4 ∈ SU (p, q), then cosh Rad(M ) = SingVal(M 1 ).
Proof. By the Cartan decomposition, there exists

K 1 , K1 ∈ U (p) and K 4 , K4 ∈ U (q) such that, if r is the radial part of M then M = M 1 M 2 M 3 M 4 = K 1 0 0 K 4   cosh r sinh r 0 sinh r cosh r 0 0 0 I q-p   K1 0 0 K4 . Therefore M 1 = K 1 (cosh r) K1 and M 1 M * 1 = K 1 (cosh 2 r)K *
1 which proves the lemma. 4.3. Iwasawa decomposition and horocyclic coordinates. It will be convenient to write matrices in SU (p, q) in another basis to make more tractable the solvable part of the Iwasawa decomposition (see, e.g., Lu [START_REF] Lu | The heat kernels of non compact symmetric spaces[END_REF], Iozzi and Morris [START_REF] Iozzi | Cartan-decomposition subgroups of SU (2, n)[END_REF] or Sawyer [START_REF] Sawyer | Spherical functions on SO(p, q)/SO(p) × SO(q)[END_REF]). We still suppose that q ≥ p. Let

P =   I p / √ 2 0 I p / √ 2 I p / √ 2 0 -I p / √ 2 0 I q-p 0   .
Then P * = P -1 and P -1 JP = J where

J =   0 0 -I p 0 I q-p 0 -I p 0 0   .
We introduce G = {P -1 M P, M ∈ SU (p, q)}, which is the set of matrices M in Sl(p + q, C) such that M * JM = J. The group G is obviously isomorphic to SU (p, q). So, we will work with G instead of SU (p, q). Let l, b, c be p × p, p × (q -p) and p × p complex matrices, respectively, where l is lower triangular and invertible, and define Notice that in general l is lower triangular and hence neither N nor S is made of upper triangular matrices. If D = {D p (r), r ∈ R p } then A = P -1 DP . The Iwasawa decomposition is G = N AK, N is the nilpotent component and S = N A. Notice that SU (p, q)/ K is isomorphic with G/K. For M ∈ G, P M P -1 is in SU (p, q) and we let Rad(M ) = Rad(P M P -1 ). In G/K, we choose as origin o = K, then Rad(M ) plays the role of a generalized distance between the cosets o = K and ξ = M K in G/K.

(4.1) S(l, b, c) =   l b cl * -1 0 I q-p b * l * -1 0 0 l * -1   .

Lemma 4.3 ([24]

). For S(l, b, c) ∈ S,

cosh Rad(S(l, b, c)) = 1 2 SingVal(l + l * -1 + cl * -1 ).
Proof. This follows immediately from Lemma 4.2 since, for S(l, b, c) ∈ G, the corresponding element in SU (p, q) can be written as

P S(l, b, c)P -1 = M 1 M 2 M 3 M 4 ,
where M 1 = (l + (I + c)l * -1 )/2.

By the Iwasawa decomposition each element ξ ∈ G/K can be written uniquely as ξ = S(l, b, c)K with S(l, b, c) ∈ S. We see that we can and will identify S and G/K. We call S(l, b, c) the horocyclic coordinates of ξ. They generalize the horocyclic-Poincaré coordinates in H q . 4.4. Brownian motion on G/K and infinite Brownian loop. The Lie algebra S of S is

S = {   l b c 0 0 b * 0 0 -l *  
, where c is skew -Hermitian, l is lower triangular with real diagonal}.

The Killing form on the Lie algebra of G allows to define a scalar product on S by

< X, Y >= 1 2 Trace(XY * ).
Then S is a real Euclidean space. Let A and N be the Lie algebras of A and N , then

S = A ⊕ N. Let X 1 , • • • , X p be
X 2 i + 2 s j=1 N 2 j -2 p i=1 ̺(X i )X i ,
where ̺ is given below by (5.1) and X i and N j are considered as left invariant vector fields. We consider the Riemannian Brownian motion {ξ t , t ≥ 0} on G/K = S. It is the process with generator ∆/2 starting from the origin o.

As in the hyperbolic case one can consider the ground state process {ξ 0 t , t ≥ 0} of this Brownian motion associated with the basic Harish Chandra spherical function ϕ 0 . By [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF]Theorem IV.4

.3], (4.2) ϕ 0 (g) = K e ̺(H(kg)) dk,
where for g ∈ G, we write H(g) for the element of the Lie algebra A of A such that g ∈ N e H(g) K in the Iwasawa decomposition G = N AK. The generator of ξ 0 is ∆ ϕ 0 /2 and it corresponds to the infinite Brownian loop on G/K (see [START_REF] Anker | The infinite Brownian loop on a symmetric space[END_REF]). By invariance of ∆ under K, the radial part of Rad(ξ t ) and Rad(ξ 0 t ) are Markov processes with values in the closed Weyl chamber A + . 4.5. Distinguished Brownian motion on S. We define Ψ 0 : G → R + by Ψ 0 (g) = e ̺(H(g)) .

Since H(gk) = H(g) for any k ∈ K, Ψ 0 is well defined on G/K. We consider the Ψ 0 -ground state process {S t , t ≥ 0} of the Brownian motion ξ on G/K. Using the identification G/K = S, we see it as a process on S (it has the same interpretation as the one given in Remark 3.9). The following definition is used in harmonic analysis (see, e.g., [5, Proposition 1.2] , [START_REF] Cowling | Weak type (1,1) estimates for heat kernel maximal functions on Lie groups[END_REF][START_REF] Cowling | Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth[END_REF]), Definition 4.4. The distinguished Brownian motion on S is the process S t , t ≥ 0.

The generator of (S t ) is

1 2 ( p i=1 X 2 i + 2 s j=1 N 2 j ).
One shows as in the hyperbolic case that (see also [1, Proof of Theorem 6.1]), Lemma 4.5. The two processes {Rad(S t ), t ≥ 0} and {Rad(ξ

(0) t ), t ≥ 0} have the same law.
The process (S t ) is a solution of a stochastic differential equation. Indeed, consider the Brownian motion (V t ) on the Lie algebra S, considered as an Euclidean space,

V t =   λ t β t κ t 0 0 β * t 0 0 -λ * t   ∈ S,
where the coefficients λ r,r t , iκ r,r t /2, 1 ≤ r ≤ p, and the real and imaginary

parts of λ r,s t / √ 2, r > s, β k,l t / √ 2, 1 ≤ k ≤ p, 1 ≤ l ≤ q -p
and κ r,s t / √ 2, 1 ≤ r < s ≤ q are standard real independent Brownian motions, λ r,s t = 0 if 1 ≤ r < s ≤ p and κ r,s t = -κ s,r t , when q ≥ r > s ≥ 1 (we use m i,j to denote the (i, j) coefficient of a matrix m).

When (X t ) is a continuous semimartingale, we use δX t for its Stratonovich differential and dX t for its Ito one (see, e.g., [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF]).

Proposition 4.6. The distinguished Brownian motion (S t ) is the solution of the following Stratonovich stochastic differential equation in the set of (p + q) × (p + q) complex matrices, δS t = S t δV t , S 0 = I p+q .

In order to compute the radial component we use a decomposition of S which is slightly different from the factorization S = N A coming from the Iwasawa factorization. We write, using notation (4.1)

S t = S(l t , b t , c t ), and
(4.3) S t = M t L t ,
where

M t =   I p b t c t 0 I q-p b * t 0 0 I p   , L t =   l t 0 0 0 I q-p 0 0 0 l * -1 t   .
Recall that the matrix l t is lower triangular. Its diagonal is the A component of S t but it also has a part of the N component. We solve the equation satisfied by (S t ). By Stratonovich calculus,

δS t = M t δL t + (δM t )L t ,
therefore, M t L t δV t = M t δL t + (δM t )L t , which implies that

δV t = L -1 t δL t + L -1 t M -1 t (δM t )L t , hence L -1 t δL t =   δλ t 0 0 0 0 0 0 0 -δλ * t   , and
L -1 t M -1 t (δM t )L t =   0 δβ t δκ t 0 0 δβ * t 0 0 0   .
We obtain that δl t = l t δλ t and

δM t = M t L t   0 δβ t δκ t 0 0 δβ * t 0 0 0   L -1 t , which gives   0 δb t δc t 0 0 δb * t 0 0 0   =   I p b t c t 0 I q-p b * t 0 0 I p     0 l t δβ t l t δκ t l * t 0 0 δβ * t l * t 0 0 0   .
This show that:

Proposition 4.7.

δl t = l t δλ t , b t = t 0 l s δβ s , c t = t 0 l s (δκ s )l * s + t 0 b s (δβ * s )l * s .
In particular, Corollary 4.8. The process (l t ) is a Brownian motion on the subgroup of Gl(p, C) consisting of lower triangular matrices with positive diagonal.

The linear equation δl t = l t δλ t is therefore easy to solve explicitly by induction on p. For instance, when p = 2, if

λ t = λ (1) t 0 λ (3) t λ (2) t , we obtain that l t = e λ (1) t 0 e λ (2) t t 0 e λ (1) s -λ (2)
s δλ

(3) s e λ (2) t .
4.6. Limit as q → +∞. We study the asymptotic behaviour of the radial part Rad(S t ) of the distinguished Brownian motion on S, using the decomposition (4.3). We first consider Ito's integral.

Lemma 4.9. Almost surely,

lim q→+∞ 1 q t 0 [ s 0 l u dβ u ](dβ * s )l * s = 0. Proof. For 1 ≤ i, j ≤ p, [ t 0 ( s 0 l u dβ u )(dβ * s )l * s ] i,j = q-p r=1 p n=1 p m=1 t 0 ( s 0 l i,n u dβ n,r u ) lj,m s d βm,r s .
We fix i, j, n, m. Conditionally on the sigma-algebra σ(l s , s ≥ 0), the random variables Proof. By Proposition 4.7,

c t = t 0 l s (δκ s )l * s + t 0 ( s 0 l u δβ u )(δβ * s )l * s .
Since the processes (l t ) and (κ t ) do not depend on q, lim q→+∞

1 q c t = lim q→+∞ 1 q t 0 ( s 0 l u δβ u )(δβ * s )l * s .
Now, recall the link between Stratonovich and Ito integral: if X and Y are continuous semimartingales,

t 0 Y δX = t 0 Y dX + 1 2 X, Y t ,
where, if X and Y are matrices, X, Y t is the matrix with (i, j) entrie

X, Y i,j t = k X i,k , Y k,j t .
Since (β t ) is independent of (l t ),

s 0 l u δβ u = s 0 l u dβ u ,
s t = q-p r=1 p k=1 t 0 l i,k s dβ k,r s , t 0 lj,k s d βk,r s t = q-p r=1 p k=1 t 0 l i,k s lj,k s d β k,r , βk,r s .
Recall that β k,r t / √ 2 is a complex Brownian motions, hence

β k,r , βk,r s = 4s, therefore q-p r=1 p k=1 t 0 l i,k s lj,k s d β k,r , βk,r s = 4(q -p) p k=1 t 0 l i,k s lj,k s ds,
and t 0 l s dβ s , t 0 dβ * s l * s t = 4(q -p) t 0 l s l * s ds.
Consequently, a.s.,

lim q→+∞ 1 q c t = lim q→+∞ 2(q -p) q t 0 l s l * s ds = 2 t 0 l s l * s ds,
and the proposition follows from Lemma 4.3 and from the equality SingVal(

t 0 l s l * s l * -1 t ds) = SingVal(l -1 t t 0 l s l * s ds).
The following is a direct consequence of the previous proposition and Lemma 4.5. Let 1 ∈ R p denote the vector 1 = (1, 1, • • • , 1). Theorem 4.11. For SU (p, q), the process η t , t ≥ 0, with values in

{(r 1 , • • • , r p ) ∈ R p , r 1 ≥ r 2 ≥ • • • ≥ r p } given by η t = SingVal(l -1 t t 0 l s l * s ds)
is a Markov process. The process Rad(ξ 0 t ) -log(2q)1, t ≥ 0, converges in distribution to log η t , t ≥ 0.

We will compute the generator of (η t ) in Theorem 5.2. 4.7. SO(p, q) and Sp(p, q). For SO(p, q) the only difference with SU (p, q) is that all the entries are real. Hence, in the preceding computation, β k,r , βk,r s = β k,r , β k,r s = 2s. Similarly, for Sp(p, q) the entries are quaternionic, therefore, in that case, β k,r , βk,r s = 8s. So we obtain, Theorem 4.12. For SO(p, q), resp. Sp(p, q), in distribution,

lim q→+∞ Rad(ξ 0 t ) -log(θq)1 = log SingVal(l -1 t t 0 l s l * s ds),
with θ = 1 for SO(p, q), resp. θ = 4 for Sp(p, q), and where l is the Brownian motion on the group of p × p lower triangular matrices with real, resp. quaternionic, entries and with positive diagonal, solution of δl t = l t δλ t . where B is a standard Brownian motion.

5.

Generator of (η t ) 5.1. Inozemtsev limit to Quantum Toda Hamiltonian. In order to compute the generator L of the process (η t ) process in Theorem 4.11 it is convenient to relate Rad(ξ 0 t ) to a Calogero Moser Sutherland model. We consider the general case of a symmetric space G/K associated with one of the groups G = SO(p, q), SU (p, q) and Sp(p, q), where K is a maximal compact subgroup. The root system is of type BC p . Recall that, if {e 1 , . . . , e p } is an orthonormal basis of a real Euclidean space a of dimension p, with dual basis (e * i ) then the positive roots Σ + of the root system BC p are given by

Σ + = {e * k , 2e * k , (1 ≤ k ≤ p), e * i + e * j , e * i -e * j , (1 ≤ i < j ≤ p)}.
We associate to each considered group a triplet m = (m 1 , m 2 , m 3 ) given by, when p ≥ 2, for SO(p, q), m = (q -p, 0, 1), for SU (p, q), m = (2(q -p), 1, 2) and for Sp(p, q), m = (4(q -p), 3, 4). When p = 1, the only difference is that m 3 = 0. The quantum Calogero-Moser-Sutherland trigonometric Hamiltonian of type BC p is (see [START_REF] Olshanetsky | Quantum integrable systems related to Lie algebras[END_REF][START_REF] Oshima | Heckman-Opdam hypergeometric functions and their specializations. In New viewpoints of representation theory and noncommutative harmonic analysis[END_REF])

H CM S =H CM S (p, q; r) = p k=1 ( ∂ 2 ∂r 2 k - m 1 (m 1 + 2m 2 -2) 4 sinh 2 r k - m 2 (m 2 -2) sinh 2 2r k ) - 1≤i<j≤p m 3 (m 3 -2)( 1 2 sinh 2 (r i -r j ) + 1 2 sinh 2 (r i + r j )
).

For

α ∈ Σ + we let m α = m 1 if α = e * k , m α = m 2 if α = 2e * k and m α = m 3 if α = e * i + e * j or α = e * i -e * j . Define (5.1) ρ = 1 2 α∈Σ + m α α, δ = α∈Σ + (e α -e -α ) mα .
The radial part ∆ R of the Laplace Beltrami operator on G/K is equal (see [18, p.268]) to

∆ R = p k=1 ∂ 2 ∂r 2 k + p k=1 ∂ ∂r k (log δ) ∂ ∂r k ,
which can be written as

∆ R = δ -1/2 • {H CM S -(ρ, ρ)} • δ 1/2
(see [START_REF] Olshanetsky | Quantum systems connected with root systems and the radial parts of Laplace operators[END_REF][START_REF] Olshanetsky | Quantum integrable systems related to Lie algebras[END_REF][START_REF] Oshima | Heckman-Opdam hypergeometric functions and their specializations. In New viewpoints of representation theory and noncommutative harmonic analysis[END_REF]). On the other hand, we define φ0 : R p → R by φ0 (r) = ϕ 0 (D p (r)) where ϕ 0 is the basic Harish Chandra function. Then φ0 (0) = 1 and ∆ R φ0 = -(ρ, ρ) φ0 . Therefore H CM S (δ 1/2 φ0 ) = 0. The generator of the radial part Rad(ξ 0 t ) of the ground state process (ξ 0 t ) associated with ϕ 0 is ∆ φ0 R /2 where

∆ φ0 R = p k=1 ∂ 2 ∂r 2 k + 2 p k=1 ∂ ∂r k (log δ 1/2 φ0 ) ∂ ∂r k = (δ 1/2 φ0 ) -1 • H CM S • (δ 1/2 φ0 ).
For SU (p, q), since m 3 = 2, we have the following, called the Inozemtsev limit ( [START_REF] Inozemtsev | The finite Toda lattices[END_REF]),

lim q→+∞ H CM S (p, q; r + log(2q)1) = p k=1 ∂ 2 ∂r 2
k -e -2r k .

Similarly, for SO(p, q),

lim q→+∞ H CM S (p, q; r + log q 1) = p k=1 ( ∂ 2 ∂r 2 k -e -2r k ) + 1≤i<j≤p 1 2 sinh 2 (r i -r j ) ,
and for Sp(p, q),

lim q→+∞ H CM S (p, q; r + log(4q)1) = p k=1 ( ∂ 2 ∂r 2 k -e -2r k ) - 1≤i<j≤p 4 sinh 2 (r i -r j )
.

We denote by H T the respective limit. Since δ 1/2 φ0 is a ground state of H CM S , it is reasonable to infer that there is a ground state Ψ of H T such that the generator L of (η t ) in Theorem 4.11 is given by

L = 1 2 Ψ -1 • H T • Ψ.
Oshima and Shimeno [START_REF] Oshima | Heckman-Opdam hypergeometric functions and their specializations. In New viewpoints of representation theory and noncommutative harmonic analysis[END_REF] goes into that direction but is not precise enough to obtain this conclusion. We will see that this hold true for SU (p, q), but that the result is quite subtle. The reason is that on the one hand, there are many ground states for H T and on the other hand that L is the generator of a process with values in a proper subcone of R p when p > 1. We don't know if this is true for S0(p, q) or Sp(p, q). 5.2. Asymptotics for SU (p, q). We compute the generator of the process η t for SU (p, q), using that in this case the spherical functions are explicitly known for all (p, q). In SO(p, q) and Sp(p, q), such an expression is not known up to now. For λ ∈ R p , the Harish Chandra spherical function ϕ

(p,q) λ of SU (p, q), is defined by (see [18, Theorem IV.4.3]), for g ∈ SU (p, q), ϕ (p,q) λ (g) = K e (λ+̺)(H(kg)) dk. Let A(p, q) = (-1) 1 2 (p(p-1) 2 2p(p-1)
p-1 j=1 {(q -p + j) p-j j!}, c(p, q) = A(p, q)(-1)

1 2 p(p-1) (2!4! • • • (2(p -1))!) -1 .
By Hoogenboom [START_REF] Hoogenboom | Spherical functions and invariant differential operators on complex Grassmann manifolds[END_REF]Theorem 3], if for all i, λ i ∈ Z, then when r ∈ R p , (

D p (r)) = A(p, q) det(ϕ (1,q-p+1) λ i (D 1 (r j )) 1≤i<j≤p (cosh 2r i -cosh 2r j )(λ 2 i -λ 2 j ) . Lemma 5.1. For r ∈ R p , ϕ (p,q) 0 (D p (r)) = c(p, q) 1≤i<j≤p (cosh 2r i -cosh 2r j ) det(M (p,q) (r)), 5.2) ϕ (p,q) λ (
where M (p,q) (r) is the p × p matrix with (i, j) coefficient given by

M (p,q) (r) i,j = d 2(j-1) dλ 2(j-1) ϕ (1,q-p+1) λ (D 1 (r i )) {λ=0} . Proof. Since; for λ ∈ R, ϕ (1,q-p+1) λ = ϕ (1,q-p+1) -λ
, the lemma follows from Hua's lemma [START_REF] Hoogenboom | Spherical functions and invariant differential operators on complex Grassmann manifolds[END_REF]Lemma 4.1].

For SU (p, q), m α = 2(q -p) when α = e * k , m α = 1 when α = 2e * k and m α = 2 when α = e * i + e * j or α = e * i -e * j . Thus, by (5.1),

δ(D p (r)) = k (2 sinh r k ) 2(q-p) (2 sinh 2r k ) i<j 4(cosh 2r i -cosh 2r j ) 2 .
On the other hand, for r ∈ R, let δ q-p+1 (r) = 2 2(q-p)+1 sinh 2(q-p) r sinh 2r.

(this is (7.1) adapted to the complex case). Then there is C(p, q) > 0 such that, for r ∈ R p ,

(δ 1/2 ϕ (p,q) 0 )(D p (r)) = C(p, q) det M (p,q) (r) p k=1 δ q-p+1 (r k ) = C(p, q) det N (p,q) (r),
where

(N (p,q) (r)) i,j = d 2(j-1) dλ 2(j-1) (δ 1/2 q-p+1 φq-p+1 λ )(r i ) {λ=0} ,
and where φq-p+1

λ (s) = ϕ (1,q-p+1) λ
(D 1 (s)) when s ∈ R. Let (see (7.4),(7.5))

a(q -p + 1) = Γ(q -p) Γ(2(q -p))2 5/2 , and g q-p+1 (λ, r) = a(q -p + 1)(δ 1/2 q-p+1 φq-p+1 λ )(r + log 2(q -p)) -K λ (e -r ). By Corollary 7.3, g q-p+1 (λ, r) and all its derivatives at λ = 0 converge to 0 as q → +∞. The generator of Rad(ξ 0 t ) is

1 2 p i=1 ∂ 2 ∂r 2 i + p i=1 ∂ ∂r i log(δ 1/2 ϕ (p,q) 0 )(D p (r)) ∂ ∂r i
.

Recall that

η t = SingVal(l -1 t t 0 l s l * s ds).
We deduce from Corollary 7.3 that:

Theorem 5.2. The generator of the process log η t , t ≥ 0, in Theorem 4.11 is 1 2

p i=1 ∂ 2 ∂r 2 i + p i=1 ∂ ∂r i log K(p) (e -r ) ∂ ∂r i
, where K(p) (e -r ) is the determinant of the p × p matrix with (i, j)coefficient d 2(j-1) dλ 2(j-1) K λ (e -r i ) {λ=0} .

Remark 5.3. The function r → K(p) (e -r ) is a ground state of the Toda Hamiltonian

p k=1 ∂ 2 ∂r 2
k -e -2r k equal to 0 on the walls {r i = r j }. Notice that p k=1 K 0 (e -r k ) is another ground state.

Remark 5.4. Exactly as in Proposition 3.13, but using (5.2), one has that, for r = log η t ,

E(e p k=1 λ k l (k,k) t |σ(η s , 0 ≤ s ≤ t)) = c p Kλ (e -r ) K(p) (e -r ) i<j (λ 2 i -λ 2 j )
, where c p = (-1)

1 2 p(p-1) (2!4! • • • (2(p - 1
))! and Kλ (e -r ) is the determinant of the matrix (K λ i (e -r j )). As in Remark 3.14, this implies that when the diagonal part of (l t ) has a drift λ, then log η t is a Markov process with generator given by 1 2

p i=1 ∂ 2 ∂r 2 i + p i=1 ∂ ∂r i
log Kλ (e -r ) ∂ ∂r i

.

Remark 5.5. After the submission of this paper, Rider and Valko [START_REF] Rider | Matrix Dufresne identities[END_REF] have posted a paper where they consider a Brownian motion M t on Gl(p, R), solution of the stochastic differential equation dM t = M t dB t + ( 1 2 + µ)M t dt where B t is the p × p matrix made of p 2 independent standard real Brownian motions. They show in particular, using a similar approach as Matsumoto and Yor, that Z t = M -1 t t 0 M s M * s ds, t ≥ 0, is a Markov process if |µ| > (p-1)/2 and describe its generator. For Gl(p, C), when µ = 0, SingVal(Z t ) = SingVal(η t ), where η t is given in the preceding remark with λ = ̺.

Series of homogeneous trees

We now consider the same question for q-adic symmetric spaces of rank one. The most important series is given by the symmetric spaces associated with Gl(2, Q q ) where q is the sequence of prime numbers which are the homogeneous trees T q . The analogue of the Brownian motion is the simple random walk. Let us consider more generally the tree T q , where q is an arbitrary integer. We will deal to this case by an elementary treatment. By definition, T q is the connected graph without cycle whose vertices have exactly q + 1 edges. We choose an origin o in this tree. The simple random walk W n , n ≥ 0, starts at o and at each step goes to one of its q+1 neighbours with uniform probability.

Let us recall some (well known) elementary facts about (W n ) ( [START_REF] Figa-Talamanca | Harmonic analysis and representation theory for groups acting on homogeneous trees[END_REF], [START_REF] Woess | Random walks on infinite graphs and groups[END_REF]). For the convenience of the reader we give the simple proofs. Let us write x ∼ y when x and y are neighbours. The probability transition of W n is

P (x, y) = 1 q + 1 , iff x ∼ y.
Let d be the standard distance on the tree. The radial part of W n is the process X n = d(o, W n ), it is a Markov chain on N with transition probability R given by R(0, 1) = 1, and if n > 0, R(n, n -1) = 1 q + 1 , R(n, n + 1) = q q + 1 .

A function f : T q → R is called radial when f (x) depends only on d(o, x).

In this case one defines f :

N → R by f (n) = f (x), when d(o, x) = n.
Let us introduce the average operator A which associates to a function f : T q → R the radial function Af : T q → R defined by

Af (x) = 1 S(o, x) y∈S(o,x) f (y),
where S(o, x) is the sphere S(o, x) = {y ∈ T q ; d(o, y) = d(o, x)}. It is easy to see that P A = AP , which implies that, Lemma 6.1. When a function f : T q → R is a λ-eigenfunction of P (i.e. P f = λf ), then Af is a radial λ-eigenfunction of P and Af is a λ-eigenfunction of R. Conversely, if g :

N → R is a λ-eigenfunction of R, then g(x) = g(d(o, x)) is a radial λ-eigenfunction of P .
Let λ > 0, g : N → R + is a λ-eigenfunction of R when g(1) = λg(0) and g(n -1) + qg(n + 1) = λ(q + 1)g(n), n ≥ 1.

One sees easily that there exists a positive solution of this equation if and only if λ ≥ 2 √ q q+1 . Hence, the principal generalized eigenvalue ̺ of R is

̺ = 2 √ q q + 1 , the associated eigenfunction φ0 is φ0 (n) = (1 + n q -1 q + 1 ) 1 q n/2 .
It is the only one if we suppose that φ0 (0) = 1. It follows from Lemma 6.1 that ̺ is also the principal generalized eigenvalue of P . The function ϕ 0 : T q → R given by ϕ 0 (x) = φ0 (d(o, x)) for x ∈ T q , is a radial ̺-eigenfunction of P . By uniqueness of φ0 , ϕ 0 is the unique radial ground state of the random walk W n , n ≥ 0, equal to 1 at o. We consider the ϕ 0 -ground state process W (0) n , n ≥ 0, on T q , defined as the Markov chain with transition probability P (0) given by, for x, y ∈ T q , P (0) (x, y) = 1 ̺φ 0 (x) P (x, y)φ 0 (y).

Its radial part is a Markov chain on N with transition probability, for m, n ∈ N,

R (0) (n, m) = 1 ̺ φ0 (n) R(n, m) φ0 (m). It is clear that lim q→+∞ R (0) (m, n) = B(m, n),
where B is transition probability of the so-called discrete Bessel(3) Markov chain on N, namely,

B(0, 1) = 1, B(n, n + 1) = n + 2 2(n + 1) , B(n, n -1) = n 2(n + 1)
.

Therefore, Proposition 6.2. When q → +∞, the radial part of the ϕ 0 -ground state process W

n of the simple random walk on the tree T q converges in distribution to the discrete Bessel(3) chain.

Our aim is to obtain a path description of this limit process in terms of the simple random walk on Z. We use the analogue of the distinguished Brownian motion. A geodesic ray ω is an infinite sequence x n , n ≥ 0, of distinct vertices in T q such that d(x n , x n+1 ) = 1 for any n ≥ 0. We fix such a geodesic ray ω = {x n , n ≥ 0} starting from x 0 = o. For any x in T q let π ω (x) be the projection of x on ω, defined by d(x, π ω (x)) = min{d(x, y), y ∈ ω} and π ω (x) ∈ ω. The height h(x) of x ∈ T q with respect to ω is

h(x) = d(x, π ω (x)) -d(o, π ω (x)), (h is a Busemann function). For n ∈ Z the horocycle H n is H n = {x ∈ T q , h(x) = n}.
As described by Cartier [START_REF] Cartier | Fonctions harmoniques sur un arbre[END_REF] (see also [START_REF] Cartwright | Random walks on the affine group of local fields and of homogeneous trees[END_REF]), T q can be viewed as an infinite genealogical tree with ω as the unique "mythical ancestor". Then H n is the n-th generation. Any vertex (individual) x in T q has q neighbours of height h(x) + 1 (his children) and one neighbour of height h(x) -1 (his parent). This implies that if W n is the simple random walk on T q , then h(W n ) is the random walk on Z with transition probability H given by, for any n ∈ Z,

H(n, n -1) = 1 q + 1 , H(n, n + 1) = q q + 1 .
A positive eigenfunction f of this kernel is a solution of

f (n -1) + qf (n + 1) = λ(q + 1)f (n), n ∈ Z, f (0) = 1,
and one sees easily that it exists if and only if

λ ≥ 2 √ q q + 1 = ̺. o H 1 H 0 H -1 H -2 ω ω ′ Figure 1. Example T 3
As a result ̺ is also the principal generalized eigenvalue (or spectral radius) of H. Moreover the associated eigenfunction is f (n) = q -n/2 , n ∈ N. This implies that the function

ϕ ω (x) = q -h(x)/2 , x ∈ T q ,
is, like ϕ 0 , a ground state of the Markov chain W n . By Lemma 6.1, Aϕ ω is a radial ground state of P . By uniqueness, this implies that (6.1) ϕ 0 = Aϕ ω . Definition 6.3. Let S n , n ≥ 0, be the ϕ ω -ground state process associated to (W n ) on T q starting from o. We denote by Q ω its probability transition.

We have, Q ω (x, y) = P (x, y) ϕ ω (y) ̺ϕ ω (x) = P (x, y) q -h(y)/2 ̺q -h(x)/2 = P (x, y) (q + 1)q h(x)-h(y)/2 2 √ q .

Therefore, when x ∼ y, since P (x, y) = 1/(q + 1),

Q ω (x, y) = 1/2q, when h(y) = h(x) + 1, 1/2, when h(y) = h(x) -1.
As in Proposition 3.8,

n ), n ≥ 0.

Proof. Let N ∈ N and F : Z N +1 → R + . For any isometry k of the tree T q which fixes o, W n , n ≥ 0, has the same law as k.W n , n ≥ 0. Hence, using

E(F (d(o, S n ), n ≤ N )) = ̺ -N E(F (d(o, W n ), n ≤ N )ϕ ω (W N )) = ̺ -N E(F (d(o, k.W n ), n ≤ N )ϕ ω (k.W N )) = ̺ -N E(F (d(o, W n ), n ≤ N )Aϕ ω (W N )) = ̺ -N E(F (d(o, W n ), n ≤ N )ϕ 0 (W N )) = E(F (d(o, W (0) n ), n ≤ N )).
We choose in the tree T q another geodesic ray ω ′ = {x -n , n ≥ 0} such that ω ∩ ω ′ = {o}. Then ωω ′ = {x n , n ∈ Z} is a two-sided geodesic. We consider the following (unoriented) graph G embedded in Z 2 (see Figure 2): the edges of G are the points with coordinates

(k + 2n, k), k ∈ Z, n ∈ N,
and the vertices are the first diagonal segments joining (k, k) and (k+1, k+1) and the segments joining (k + 2n, k) and (k + 2n + 1, k -1) for n ≥ 0. It is obtained from the tree by gathering siblings (with one exception on each vertex of ωω ′ ). Let Ψ : T q → G defined as follows: for x ∈ T q , Ψ(x) = (-h(x) + 2d(x, ω ′ ω), -h(x)).

Then Ψ(S n ), n ≥ 0, is the nearest neighbour Markov chain on the graph G with probability transition P G given by

P G ((k, k), (k -1, k -1)) = 1/2q, P G ((k, k), (k + 1, k + 1)) = 1/2, P G ((k, k), (k + 1, k -1) = (q -1)/2q,
and, for n > 0 and ε = 1 or -1,

P G ((k + 2n, k), (k + 2n -ε, k + ε)) = 1/2.
When q → +∞, the Markov chain Ψ(S n ) converges to the Markov chain τ n , n ∈ N, on the graph G with transition probability Q given by

Q((k, k), (k -1, k -1)) = 0, Q((k, k), (k + 1, k + 1)) = 1/2, Q((k, k), (k + 1, k -1) = 1/2,
and for n > 0 and ε = 1 or -1,

Q((k + 2n, k), (k + 2n -ε, k + ε)) = 1/2.
When the chain τ n starts from (0, 0) it cannot reach (-1, -1) therefore it lives on the subgraph G of G described in Figure 3. It is the one which appears in the discrete time Pitman theorem (see Figure 8 in Biane [START_REF] Biane | Introduction to random walks on noncommutative spaces[END_REF]). It has the following simple description in terms of the simple symmetric random walk Σ n , n ≥ 0, on Z.

Let Σ n = ε 1 + • • • + ε n where the (ε n ) are i.i.d. random variables such that P(ε n = 1) = P(ε n = -1) = 1/2 and M n = max{Σ k , k ≤ n}.
The Markov chain τ n on G starting from (0, 0) is, in xOy coordinates,

τ n = (2M n -Σ n , Σ n ),
(see Figure 3). The radial component of S n is d(o, S n ) = δ((0, 0), Ψ(S n )) for the graph distance δ on G. Since δ((0, 0), (a, b)) = a, we obtain using Lemma 6.4 that, Theorem 6.5. The limit as q → +∞ of the processes d(o, S n ), n ≥ 0, and d(o, W where Σ n is the simple symmetric random walk on Z.

By Proposiiton 6.2, we recover the following theorem of Pitman.

Corollary 6.6 ( [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF]). The process 2 max 0≤k≤n Σ k -Σ n , n ≥ 0, is the discrete Bessel(3) Markov chain.

Appendix : Asymptotics of spherical functions in rank one

We describe the needed asymptotic behaviour of spherical functions on SO(1, q), SU (1, q) and Sp(1, q) when q → +∞. We adapt Shimeno [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF] to this setting (which only considers the real split case) and Oshima and Shimeno [START_REF] Oshima | Heckman-Opdam hypergeometric functions and their specializations. In New viewpoints of representation theory and noncommutative harmonic analysis[END_REF]. The advantage of this approach is that it is adapted to higher rank cases. The details of the computations are quite long but straightforward. Therefore we only indicate the main points of the proof where they differ from [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF].

We adapt the notions of Section 5.1 to the rank one case. In this case there are at most two roots, α and 2α and we may suppose that α(r) = r when r ∈ a = R. Their multiplicity are (m α , m 2α ) = (q -1, 0) for SO(1, q), (2(q -1), 1) for SU (1, q) and (4(q -1), 3) for Sp(1, q). Let ̺ q = 1 2 (m α + 2m 2α ), and (7.1) δ q (r) = (e r -e -r ) mα (e 2r -e -2r ) m 2α .

Then (see (5.1))

H CM S = δ 1/2 q • {∆ R + ρ 2 q } • δ -1/2 q = d 2 dr 2 - m α (m α + 2m 2α -2) sinh 2 r - m 2α (m 2α -2) sinh 2 2r .
For λ ∈ C, the spherical function ϕ λ satisfies ∆ R φλ = (λ 2 -̺ 2 q ) φλ , where φλ (r) = ϕ λ (D 1 (r)), r ∈ R, therefore, H CM S (δ 1/2 q φλ ) = λ 2 δ 1/2 q φλ . There exists a unique function Ψ CM S (λ, q, r), r ∈ R, of the form (7.2) Ψ CM S (λ, q, r) = n∈N b n (λ, q)e (λ-n)r , b 0 (λ, q) = 1, such that H CM S Ψ CM S = λ 2 Ψ CM S , (see [38, (17)]). When q → +∞, lim H CM S (r + log m α ) = H T , where H T is the Toda type Hamiltonian

H T = d 2 dr 2 -e -2r .
There is also a unique function Ψ T (λ, r), r ∈ R, of the form such that H T Ψ T = λ 2 Ψ T , ( [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF], notice that this is the function denoted Ψ T (-λ, -r) in [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF]). Proof. The proof is similar to the one of Proposition 1 in [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF]. Let (7.4) a(q) = Γ(m α /2) 2 Γ(m α )2 1+3m 2α /2 , and (7.5) g q (λ, r) = a(q)(δ 1/2 q φλ )(log m α + r) -K λ (e -r ). Proposition 7.2. Uniformly on λ in a small neighborhood of 0 in C, λg q (λ, r) and its derivatives in λ converge to 0 as q → +∞.

Proof. Using the Harish Chandra spherical function expansion of ϕ λ (see [18, Theorems IV.5.5 and IV.6.4] or [START_REF] Gangolli | Harmonic Analysis of Spherical Functions on Real Reductive Groups[END_REF]), when λ is for instance in the ball {λ ∈ C; |λ| ≤ 1/4} and λ = 0, one has δ 1/2 q (r) φλ (r) = c(λ)Ψ CM S (λ, q, r) + c(-λ)Ψ CM S (-λ, q, r), where c(λ) = 2 Ψ CM S (λ, q, r + log m α ) = Γ(λ)2 λ-1 Ψ T (λ, r).

Hence lim q→+∞ a(q)(δ 1/2 q φλ )(r+log m α ) = Γ(λ)2 λ-1 Ψ T (λ, r)+Γ(-λ)2 -λ-1 Ψ T (-λ, r).

The limit can be expressed in terms of the Whittaker function for Sl(2, R) as in [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF]Theorem 3]. Due to the relation between Whittaker and Macdonald functions (see, e.g., Bump [START_REF] Bump | Automorphic Forms and Representations[END_REF]), one finds that this limit is K λ (e -r ), thus g q (λ, r) tends to 0 when q → +∞. Now we remark that, for r fixed, the functions λg q (λ, r) are analytic functions in λ in the ball {λ ∈ C; |λ| ≤ 1/4} which are uniformly bounded in q by the computations of [START_REF] Shimeno | A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems[END_REF]Proposition 1]. Notice that we have to multiply by λ to avoid the singularity at 0 of the Harish Chandra c function. The uniform convergence of λg q (λ, r) and its derivatives in λ thus follows from Montel's theorem.

Corollary 7.3. As q → +∞, g q (λ, r) and all its derivatives at λ = 0 converge to 0.

Proof. This follows from the proposition and the fact that d n dλ n g q (λ, r) {λ=0} = 1 n + 1 d n+1 dλ n+1 (λg q (λ, r)) {λ=0} .

e 0 e

 0 Bs dβ (k) s ) 2 |σ(B r , r ≥ 0)) = t 2Bs ds.

  r is unique and is called the radial part of M . We let r = Rad(M ). Recall that, Definition 4.1. Let N be a p×p square matrix, the vector of singular values of N is SingVal(N ) = (σ 1 , • • • , σ p ) ∈ A + , where the σ i 's are the square roots of the eigenvalues of N N * written in decreasing order.

Lemma

LetS=

  {S(l, b, c); l is lower triangular with positive diagonal, c + c * = bb * }, A = {S(l, 0, 0) ∈ S; l is diagonal with positive diagonal}, N = {S(l, b, c) ∈ S; the diagonal elements of l are equal to 1}, K = P -1 KP.

  an orthonormal basis of A and N 1 , • • • , N s be an orthonormal (real) basis of N adapted to the root space decomposition. In horospherical coordinates the Laplace Beltrami operator on G/K = S is (e.g. [5, proof of Proposition 2.2],[START_REF] Cowling | Weak type (1,1) estimates for heat kernel maximal functions on Lie groups[END_REF],[17, p.105])

  u dβ n,r u ) lj,m s d βm,r s for r = 1, 2, • • • , are independent with the same law, and with expectation equal to 0 since they are martingales. Therefore, by the law of large numbers, u dβ n,r u ) lj,m s d βm,r s converges a.s. to 0 when q → +∞, which proves the lemma.Proposition 4.10. Let S t , t ≥ 0, be the distinguished Brownian motion. Then, a.s.,

Corollary 4 . 13 . 0 e

 4130 In rank one, i.e. p = 1, lim q→+∞ Rad(ξ 0 t ) -log(θq)1 = log t 2Bs-Bt ds,

Lemma 6 . 4 .

 64 The process d(o, S n ), n ≥ 0, has the same law as d(o, W

Figure 2 .

 2 Figure 2. The graph G

  n ≥ 0, have the same law as2 max 0≤k≤n Σ k -Σ n , n ≥ 0,

Figure 3 .

 3 Figure 3. Pitman's walk on the graph G

(7. 3 )

 3 Ψ T (λ, r) = n∈N b n (λ)e (λ-n)r , b 0 (λ) = 1,

Lemma 7 . 1 .Ψ

 71 If λ ∈ a * C and 2λ ∈ Z * , then lim CM S (λ, q, r + log m α ) = Ψ T (λ, r)

1 2

 1 mα+m 2α -λ Γ( 1 2 (m α + m 2α + 1

  3.14. It is straightforward to deduce from this proposition, Theorem 3.10 and Cameron and Martin's theorem that, as proved by Matsumoto -Yor[START_REF] Matsumoto | A version of Pitman's 2M -X theorem for geometric Brownian motions[END_REF], when B

	log	t 0 e 2B (λ) s -B t (λ)	(λ) t ds, t ≥ 0, is a Markov process on R with generator = B t + λt is a Brownian motion with drift λ, then 1 2 d 2 dr 2 + ( d dr log K λ (e -r )) d dr .

  and

	0	t	[	0	s	l u δβ u ](δβ * s )l * s =	0	t	[	0	s	l u dβ u ](dβ * s )l * s +	1 2	0	t	l s dβ s ,	0	t	dβ * s l * s t ,
	so it follows from the preceding lemma that
						lim q→+∞	1 q	c t =	1 2		lim q→+∞		0	t	l s dβ s ,	0	t	dβ * s l * s t .
	On the other hand, for 1 ≤ i, j ≤ p, t 0 l s dβ s , t 0 dβ * i,j s l * s t
						=	q-p r=1	(	0	t	l s dβ s ) i,r , (	0	t	dβ * s l * s ) r,j	t
						=	q-p r=1	p k=1		p l=1	0	t	l i,k s dβ k,r s ,	0	t	lj,l

s d βl,r

Mathematics Subject Classification. Primary 60J65, 60J27; Secondary 51M10, 53C35.1