Infinite-dimensional $q$-Jacobi Markov processes

Abstract

The classical Jacobi polynomials on the interval $[-1,1]$ are eigenfunctions of a second-order differential operator. It is well known that this operator generates a diffusion process on $[-1,1]$. Further, this fact admits an extension to $N$ dimensions (Demni, 2010; Remling–Rösler, 2011) leading to a $3$-parameter family of diffusion processes $X_N$ on the space of $N$-particle configurations in $[-1,1]$. The generators of the processes $X_N$ are related to Heckman–Opdam’s Jacobi polynomials attached to the root system ${\mathrm{BC}}_N$. The first result of the paper shows that the processes $X_N$ have a $q$-analog, the $N$-dimensional $q$-Jacobi processes. These are Feller Markov processes related to the $N$-variate symmetric big $q$-Jacobi polynomials. The latter polynomials were introduced and studied by Stokman (1997) and Stokman–Koornwinder (1997); they depend on two Macdonald parameters $(q,t)$ and $4$ extra continuous parameters. The $N$-dimensional $q$-Jacobi processes are still defined on a space of $N$-particle configurations, except that now the particles live not on $[-1,1]$ but on certain one-dimensional $q$-grids. The second result (the main one) asserts that the $N$-dimensional $q$-Jacobi processes survive a limit transition as $N$ goes to infinity and two of the extra parameters vary together with $N$ in a certain way. In the limit, one obtains a family of Feller Markov processes which are infinite dimensional in the sense that they live on configurations with infinitely many particles. The proof uses a lifting of the multivariate big $q$-Jacobi polynomials to the algebra of symmetric functions – a construction that does not hold for the Heckman–Opdam Jacobi polynomials. Note also that the large-$N$ limit transition is carried out without any space scaling, which would be impossible in the continuous case.