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Interacting diffusions on positive definite matrices. (English) Zbl 1479.60169

This paper considers systems of Brownian particles in the space of positive definite matrices (non-commutative processes), which evolve independently apart from some simple interactions. Some examples are given which have an integrable structure and are related to \(K\)-Bessel functions of matrix argument and multivariate generalizations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

MSC:

60J65 Brownian motion
60K35 Interacting random processes; statistical mechanics type models; percolation theory
22E30 Analysis on real and complex Lie groups
43A85 Harmonic analysis on homogeneous spaces
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[1] Assiotis, T.; O’Connell, N.; Warren, J.; Donati-Martin, C.; Lejay, A.; Rouault, A., Interlacing diffusions, Séminaire de Probabilités L. Lecture Notes in Mathematics (2019), Cham: Springer, Cham · Zbl 1452.60049
[2] Baudoin, F., Further exponential generalization of Pitman’s \(2M-X\) theorem, Electron. Commun. Probab., 7, 37-46 (2002) · Zbl 1008.60088
[3] Baudoin, F.; O’Connell, N., Exponential functionals of Brownian motion and class-one Whittaker functions, Ann. Inst. H. Poincaré Probab. Stat., 47, 1096-1120 (2011) · Zbl 1269.60066 · doi:10.1214/10-AIHP401
[4] Biane, P.; Bougerol, P.; O’Connell, N., Littelmann paths and Brownian paths, Duke Math. J., 130, 127-167 (2005) · Zbl 1161.60330 · doi:10.1215/S0012-7094-05-13014-9
[5] Borodin, A.; Corwin, I., Macdonald processes, Probab. Theory Relat. Fields, 158, 225-400 (2014) · Zbl 1291.82077 · doi:10.1007/s00440-013-0482-3
[6] Borodin, A.; Corwin, I.; Ferrari, P., Free energy fluctuations for directed polymers in random media in \(1+1\) dimension, Commun. Pure Appl. Math., 67, 1129-1214 (2014) · Zbl 1295.82035
[7] Bougerol, P.: The Matsumoto and Yor process and infinite dimensional hyperbolic space. In: In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Springer (2015) · Zbl 1334.60172
[8] Ph. Bougerol, private communication
[9] Bougerol, P.; Jeulin, T., Paths in Weyl chambers and random matrices, Probab Theory Relat. Fields, 124, 517-543 (2002) · Zbl 1020.15024 · doi:10.1007/s004400200221
[10] Bruschi, M.; Manakov, SV; Ragnisco, O.; Levi, D., The nonabelian Toda lattice: discrete analogue of the matrix Schrödinger spectral problem, J. Math. Phys., 21, 2749 (1980) · Zbl 0451.35054 · doi:10.1063/1.524393
[11] Bueno, MI; Furtado, S.; Johnson, CR, Congruence of Hermitian matrices by Hermitian matrices, Linear Algebra Appl., 425, 63-76 (2007) · Zbl 1126.15011 · doi:10.1016/j.laa.2007.03.016
[12] Butler, RW; Wood, AT, Laplace approximation for Bessel functions of matrix argument, J. Comput. Appl. Math., 155, 359-382 (2003) · Zbl 1027.65033 · doi:10.1016/S0377-0427(02)00874-9
[13] Cerenzia, M.: A path property of Dyson gaps, Plancherel measures for \(Sp(\infty )\), and random surface growth (2015). arXiv:1506.08742
[14] Corwin, I.; O’Connell, N.; Seppäläinen, T.; Zygouras, N., Tropical combinatorics and Whittaker functions, Duke Math. J., 163, 513-563 (2014) · Zbl 1288.82022 · doi:10.1215/00127094-2410289
[15] De Bruijn, N., On some multiple integrals involving determinants, J. Indian Math. Soc. New Ser., 19, 133-151 (1955) · Zbl 0068.24904
[16] Dufresne, D., The integral of geometric Brownian motion, Adv. Appl. Probab., 33, 223-241 (2001) · Zbl 0980.60103 · doi:10.1017/S0001867800010715
[17] Dynkin, EB, Non-negative eigenfunctions of the Laplace-Beltrami operator and Brownian motion in certain symmetric spaces, Dokl. Akad. Nauk SSSR, 141, 288-291 (1961) · Zbl 0116.36106
[18] Ethier, SN; Kurtz, TG, Markov Processes: Characterization and Convergence (1986), New York: Wiley, New York · Zbl 0592.60049 · doi:10.1002/9780470316658
[19] Fitzgerald, W.; Warren, J., Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions, Probab Theory Relat. Fields, 178, 121-171 (2020) · Zbl 1471.60130 · doi:10.1007/s00440-020-00972-z
[20] Grabsch, A.: Théorie des matrices aléatoires en physique statistique: théorie quantique de la diffusion et systèmes désordonnés. PhD thesis, Université Paris-Saclay (2018)
[21] Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss-Givental representation of quantum Toda chain wave equation. Int. Math. Res. Not. 1-23 (2006) · Zbl 1142.17019
[22] Herz, CS, Bessel functions of matrix argument, Ann. Math., 61, 474-523 (1955) · Zbl 0066.32002 · doi:10.2307/1969810
[23] Imamura, T.; Sasamoto, T., Determinantal structures in the O’Connell-Yor directed random polymer model, J. Stat. Phys., 163, 675-713 (2016) · Zbl 1342.82163 · doi:10.1007/s10955-016-1492-1
[24] Kelly, FP, Markovian functions of a Markov chain, Sankya Ser A, 44, 372-379 (1982) · Zbl 0586.60057
[25] Kemeny, JG; Snell, JL, Finite Markov Chains (1960), Princeton: Van Nostrand, Princeton · Zbl 0089.13704
[26] Kurtz, TG, Martingale problems for conditional distributions of Markov processes, Electron. J. Probab., 3, 1-29 (1998) · Zbl 0907.60065 · doi:10.1214/EJP.v3-31
[27] Kurtz, TG; Crisan, D., Equivalence of stochastic equations and martingale problems, Stochastic Analysis 2010 (2011), New York: Springer, New York · Zbl 1236.60073
[28] Liechty, K., Nguyen, G.B., Remenik, D.: Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy-Widom GOE distribution. arXiv:2009.07781
[29] Matheny, D., Johnson, C.R.: Congruence of Hermitian matrices by Hermitian matrices. William and Mary NSF-REU report (2005)
[30] Matsumoto, H.; Yor, M., A version of Pitman’s \(2M-X\) theorem for geometric Brownian motions, C. R. Acad. Sci. Paris, 328, 1067-1074 (1999) · Zbl 0936.60076
[31] Nguyen, GB; Remenik, D., Non-intersecting Brownian bridges and the Laguerre orthogonal ensemble, Ann. Inst. H. Poincaré Probab. Stat., 53, 2005-2029 (2017) · Zbl 1382.60122 · doi:10.1214/16-AIHP781
[32] Nomura, T., Algebraically independent generators of invariant differential operators on a symmetric cone, J. Reine Angew. Math., 400, 122-133 (1989) · Zbl 0667.43007
[33] Norris, JR; Rogers, LCG; Williams, D., Brownian motions of ellipsoids, Trans. Am. Math. Soc., 294, 757-765 (1986) · Zbl 0613.60072 · doi:10.1090/S0002-9947-1986-0825735-5
[34] O’Connell, N., Directed polymers and the quantum Toda lattice, Ann. Probab., 40, 437-458 (2012) · Zbl 1245.82091 · doi:10.1214/10-AOP632
[35] O’Connell, N., Geometric RSK and the Toda lattice, Illinois J. Math., 57, 883-918 (2013) · Zbl 1325.37047 · doi:10.1215/ijm/1415023516
[36] O’Connell, N.: Whittaker functions and related stochastic processes. In: Random Matrices, Interacting Particle Systems and Integrable Systems, MSRI, vol. 65 (2014) · Zbl 1338.60203
[37] O’Connell, N.: Stochastic Bäcklund transformations. In: In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Springer (2015) · Zbl 1338.37101
[38] O’Connell, N.; Seppäläinen, T.; Zygouras, N., Geometric RSK correspondence, Whittaker functions and symmetrized random polymers, Invent. Math., 197, 361-416 (2014) · Zbl 1298.05323 · doi:10.1007/s00222-013-0485-9
[39] O’Connell, N.; Yor, M., Brownian analogues of Burke’s theorem, Stoch. Process. Appl., 96, 285-304 (2001) · Zbl 1058.60078 · doi:10.1016/S0304-4149(01)00119-3
[40] Popowicz, Z., Some remarks about the lattice chiral field, Phys. Lett. A, 81, 235-236 (1981) · doi:10.1016/0375-9601(81)90249-8
[41] Popowicz, Z., The generalized non-abelian Toda lattice, Z. Phys. C Part. Fields, 19, 79-81 (1983) · doi:10.1007/BF01572340
[42] Rider, B., Valkó, B.: Matrix Dufresne identities. Int. Math. Res. Not. 174-218 (2016) · Zbl 1344.60070
[43] Rogers, LCG; Pitman, JW, Markov functions, Ann. Prob., 9, 573-582 (1981) · Zbl 0466.60070
[44] Seppäläinen, T.; Valkó, B., Bounds for scaling exponents for a \(1+1\) dimensional directed polymer in a Brownian environment, Alea, 7, 451-476 (2010) · Zbl 1276.60117
[45] Spohn, H.: KPZ scaling theory and the semidiscrete directed polymer model. In: Random matrices, interacting particle systems and integrable systems, MSRI, vol 65 (2014) · Zbl 1329.82149
[46] Stade, E., Archimedean \(L\)-factors on \(GL(n) \times GL(n)\) and generalized Barnes integrals, Israel J. Math., 127, 201-219 (2002) · Zbl 1032.11020
[47] Terras, A., Harmonic Analysis on Symmetric Spaces (2015), New York: Springer, New York · Zbl 0574.10029
[48] Wang, Z-L; Li, S-H, BKP hierarchy and Pfaffian point process, Nucl. Phys. B, 939, 447-464 (2019) · Zbl 1409.37068 · doi:10.1016/j.nuclphysb.2018.12.028
[49] Wonham, W., On a matrix Riccati equation of stochastic control, SIAM J. Control, 6, 681-697 (1968) · Zbl 0182.20803 · doi:10.1137/0306044
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