Elementary symmetric polynomials and martingales for Heckman-Opdam processes

Author:

Rösler Margit,Voit Michael

Abstract

We consider the generators L k L_k of Heckman-Opdam diffusion processes in the compact and non-compact case in N N dimensions for root systems of type A A and B B , with a multiplicity function of the form k = κ k 0 k=\kappa k_0 with some fixed value k 0 k_0 and a varying constant κ [ 0 , [ \kappa \in \,[0,\infty [ . Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the L k L_k for all κ ] 0 , [ \kappa \in \,]0,\infty [ . This leads to martingales associated with the Heckman-Opdam diffusions ( X t , 1 , , X t , N ) t 0 (X_{t,1},\ldots ,X_{t,N})_{t\ge 0} . As our results extend to the freezing case κ = \kappa =\infty with a deterministic limit after some renormalization, we find formulas for the expectations E ( j = 1 N ( y X t , j ) ) , \mathbb E(\prod _{j=1}^N(y-X_{t,j})), y C y\in \mathbb C .

Publisher

American Mathematical Society

Reference28 articles.

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