Exponential functionals of brownian motion and class-one Whittaker functions
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1096-1120.

Nous étudions certaines fonctionelles d'un mouvement Brownien avec dérive dans ℝn qui sont définies par une collection de fonctionnelles linéaires. Nous donnons une caractérisation de la transformée de Laplace de leur loi jointe comme l'unique solution bornée, à une constante près d'une équation aux dérivées partielles de type Schrödinger. Nous déduisons une équation similaire pour la densité. Nous caractérisons ensuite toutes les diffusions qui peuvent être interprétées comme ayant la loi d'un mouvement Brownien avec dérive conditionné par la loi de ses fonctionelles exponentielles. Dans le cas où la famille des fonctionelles est un ensemble de racines simples, la transformée de Laplace de la densité jointe des fonctionnelles exponentielles correspondantes peut être exprimée en termes d'une fonction de Whittaker de classe 1 associée au système. Dans ce cadre, nous établissons quelques propriétés du processus de diffusion correspondant.

We consider exponential functionals of a brownian motion with drift in ℝn, defined via a collection of linear functionals. We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrödinger-type partial differential equation. We derive a similar equation for the probability density. We then characterise all diffusions which can be interpreted as having the law of the brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (class-one) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes.

DOI : https://doi.org/10.1214/10-AIHP401
Classification : 60J65,  60J55,  37K10,  22E27
Mots clés : conditioned brownian motion, quantum Toda lattice
@article{AIHPB_2011__47_4_1096_0,
     author = {Baudoin, Fabrice and O{\textquoteright}Connell, Neil},
     title = {Exponential functionals of brownian motion and class-one Whittaker functions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1096--1120},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {4},
     year = {2011},
     doi = {10.1214/10-AIHP401},
     zbl = {1269.60066},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/10-AIHP401/}
}
Baudoin, Fabrice; O’Connell, Neil. Exponential functionals of brownian motion and class-one Whittaker functions. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1096-1120. doi : 10.1214/10-AIHP401. http://archive.numdam.org/articles/10.1214/10-AIHP401/

[1] L. Alili, H. Matsumoto and T. Shiraishi. On a triplet of exponential Brownian functionals. In Séminaire de probabilités de Strasbourg, XXXV 396-415. Lecture Notes in Math. 1755. Springer, Berlin, 2001. | Numdam | MR 1837300 | Zbl 0981.60080

[2] F. Baudoin. Further exponential generalization of Pitman's 2M−X theorem. Electron. Comm. Probab. 7 (2002) 37-46 (electronic). | MR 1887172 | Zbl 1008.60088

[3] F. Baudoin. Conditioned stochastic differential equations: Theory, examples and applications to finance. Stochastic Process. Appl. 100 (2002) 109-145. | MR 1919610 | Zbl 1058.60040

[4] P. Biane, P. Bougerol and N. O'Connell. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005) 127-167. | MR 2176549 | Zbl 1161.60330

[5] P. Bougerol and T. Jeulin. Paths in Weyl chambers and random matrices. Probab. Theory Related Fields 124 (2002) 517-543. | MR 1942321 | Zbl 1020.15024

[6] D. Bump. Automorphic Forms on GL(3, ℝ). Lecture Notes in Math. 1083. Springer, Berlin, 1984. | MR 765698 | Zbl 0543.22005

[7] D. Bump and J. Huntley. Unramified Whittaker functions for GL(3, ℝ). J. Anal. Math. 65 (1995) 19-44. | MR 1335367 | Zbl 0839.22016

[8] D. Dufresne. The integral of geometric Brownian motion. Adv. in Appl. Probab. 33 (2001) 223-241. | MR 1825324 | Zbl 0980.60103

[9] R. Ghomrasni. On distribution associated with the generalized Levy's stochastic area formula. Studia Sci. Math. Hungar. 41 (2004) 93-100. | MR 2082064 | Zbl 1059.60049

[10] S. G. Gindikin and F. I. Karpelevich. The Plancherel measure for Riemannian symmetric spaces with non-positive curvature. Dokl. Akad. Nauk USSR 145 (1962) 252-255. | MR 150239 | Zbl 0156.03201

[11] A. Givental. Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. In Topics in Singularity Theory 103-115. AMS Transl. Ser. 2 180. AMS, Providence, RI, 1997. | MR 1767115 | Zbl 0895.32006

[12] C. Grosche. The path integral on the Poincaré upper half-plane with a magnetic field and for the Morse potential. Ann. Phys. 187 (1988) 110-134. | MR 969177 | Zbl 0652.58042

[13] M. Hashizume. Whittaker functions on semisimple Lie groups. Hiroshima Math. J. 12 (1982) 259-293. | MR 665496 | Zbl 0524.43005

[14] N. Ikeda and H. Matsumoto. Brownian motion on the Hyperbolic plane and Selberg trace formula. J. Funct. Anal. 163 (1999) 63-110. | MR 1682843 | Zbl 0928.60063

[15] H. Jacquet. Fonctions de Whittaker associées aux groupes de Chevalley. Bull. Soc. Math. France 95 (1967) 243-309. | Numdam | MR 271275 | Zbl 0155.05901

[16] S. Kharchev and D. Lebedev. Integral representations for the eigenfunctions of a quantum periodic Toda chain. Lett. Math. Phys. 50 (1999) 53-77. | MR 1751619 | Zbl 0970.37056

[17] B. Kostant. Quantisation and representation theory. In Representation Theory of Lie Groups, Proc. SRC/LMS Research Symposium, Oxford 1977 287-316. LMS Lecture Notes 34. Cambridge Univ. Press, Cambridge, 1977. | Zbl 0474.58010

[18] N. N. Lebedev. Special Functions and Their Applications. Dover, New York, 1972. | MR 350075 | Zbl 0271.33001

[19] H. Matsumoto and M. Yor. A version of Pitman's 2M−X theorem for geometric Brownian motions. C. R. Acad. Sci. Paris Sér. 1 328 (1999) 1067-1074. | MR 1696208 | Zbl 0936.60076

[20] H. Matsumoto and M. Yor. Exponential functionals of Brownian motion, I: Probability laws at a fixed time. Probab. Surv. 2 (2005) 312-347. | MR 2203675 | Zbl 1189.60150

[21] H. Matsumoto and M. Yor. A relationship between Brownian motions with opposite drifts. Osaka J. Math. 38 (2001) 383-398. | MR 1833628 | Zbl 0981.60078

[22] N. O'Connell. Directed polymers and the quantum Toda lattice. Ann. Probab. To appear, 2011. Available at arXiv:0910.0069.

[23] N. O'Connell and M. Yor. Brownian analogues of Burke's theorem. Stochastic Process. Appl. 96 (2001) 285-304. | MR 1865759 | Zbl 1058.60078

[24] N. O'Connell and M. Yor. A representation for non-colliding random walks. Electron. Comm. Probab. 7 (2002) 1-12. | MR 1887169 | Zbl 1037.15019

[25] J. W. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 (1975) 511-526. | MR 375485 | Zbl 0332.60055

[26] L. C. G. Rogers and J. Pitman. Markov functions. Ann. Probab. 9 (1981) 573-582. | MR 624684 | Zbl 0466.60070

[27] M. Semenov-Tian-Shansky. Quantisation of open Toda lattices. In Dynamical Systems VII: Integrable Systems, Nonholonomic Dynamical Systems 226-259. V. I. Arnol'd and S. P. Novikov (Eds). Encyclopaedia of Mathematical Sciences 16. Springer, Berlin, 1994. | MR 1256257 | Zbl 0795.00013

[28] E. Stade. Poincaré series for GL(3, ℝ)-Whittaker functions. Duke Math. J. 58 (1989) 695-729. | MR 1016442 | Zbl 0699.10041

[29] A. Vinogradov and L. Takhtadzhyan. Theory of Eisenstein series for the group SL(3, ℝ) and its application to a binary problem. J. Soviet Math. 18 (1982) 293-324. | Zbl 0476.10024