Some Martingales Associated With Multivariate Bessel Processes
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SOME MARTINGALES ASSOCIATED WITH MULTIVARIATE BESSEL PROCESSES M. KORNYIK1,2,† , M. VOIT3,∗ and J. WOERNER3 1
Department of Probability Theory and Statistics, E¨ otv¨ os Lor´ and University, P´ azm´ any P´ eter s´ et´ any 1/C, H-1117 Budapest, Hungary
2 Department of Quantum Optics and Quantum Information, Wigner Research Centre for Physics, Konkoly-Thege Mikl´ os u ´t 29-33, H-1121 Budapest, Hungary e-mail: [email protected] 3
Fakult¨ at Mathematik, Technische Universit¨ at Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany e-mails: [email protected], [email protected] (Received February 6, 2020; revised July 16, 2020; accepted July 20, 2020)
Abstract. We study Bessel processes on Weyl chambers of types A and B on RN . Using elementary symmetric functions, we present several space-timeharmonic functions and thus martingales for these processes (Xt )t≥0 which are independent from one parameter of these processes. As a consequence, pt (y) := i E( N i=1 (y − Xt )) can be expressed via classical orthogonal polynomials. Such formulas on characteristic polynomials admit interpretations in random matrix theory where they are partially known by Diaconis, Forrester, and Gamburd.
1. Introduction Interacting Calogero–Moser–Sutherland particle models on R with N particles can be described via Bessel processes (Xt )t≥0 associated to root systems; see e.g. [1–3,6,8,11,19–21]. These processes are classified via root systems and finitely many multiplicity parameters which control the interaction. In this paper, we study the root systems AN −1 and BN . ∗ Corresponding
author. first author has been supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity to visit Dortmund for the preparation of this paper, and also by Project no. ED 18-1-2019-0030 (Application domain specific highly reliable IT solutions subprogramme) which has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme funding scheme. Key words and phrases: Interacting particle system, Calogero–Moser–Sutherland model, zeros of Hermite polynomials, zeros of Laguerre polynomials, β-Hermite ensemble, β-Laguerre ensemble. Mathematics Subject Classification: 60F15, 60F05, 60J60, 60B20, 60H20, 70F10, 82C22, 33C67. † The
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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M. M. KORNYIK, KORNYIK, M. VOIT and J. WOERNER
In the case AN −1 , the model has a multiplicity parameter β ∈ [0, ∞[, and the associated diffusions (Xt )t≥0 live in the closed Weyl chamber A := x ∈ RN : x1 ≥ x2 ≥ . . . ≥ xN CN and satisfy the stochastic differential equation (SDE) (1.1)
dXt,i = dBt,i + β
j:j=i
dt Xt,i − Xt,j
(i = 1, . . . , N ),
where (Bt,1 , . . . , Bt,N )t≥0 denotes a standard N -dimensional Brownian moA . It is welltion, and where all paths are reflected on the boundary of CN known (see Lemma 3.4, Corollary 6.6, and Proposi
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