Infinite-dimensional $q$-Jacobi Markov processes

TL;DR

The article builds a q-deformed, infinite-dimensional generalization of Jacobi-type Markov dynamics by leveraging the algebra of symmetric functions and big $q$-Jacobi symmetric functions. It introduces the diagonal operator $\mathcal{D}_\infty$ with eigenbasis $\{\Phi_\lambda(-;q,t;\alpha,\beta;\gamma,\delta)\}$ and eigenvalues $v_\lambda$, then proves that its closure generates a strongly continuous contraction semigroup on a compact configuration space $\widetilde{\Omega}(q,t;\alpha,\beta)$ with a unique stationary measure. Finite-$N$ dynamics are constructed via $N$-variate big $q$-Jacobi polynomials and then connected to the infinite-dimensional limit through stochastic links and the intertwinings formalism, yielding a family of Feller Markov processes living on configurations with infinitely many particles, without any space scaling. The results illuminate long-range interacting, infinite-particle dynamics tied to Macdonald and Koornwinder/corepresentations, and provide a rigorous probabilistic realization of q-Jacobi limits in the spirit of Demni and Remling–Rösler, but in a totally disconnected limit setting.

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 $BC_N$. 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 later 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, only 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's 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.

Theorems & Definitions (47)

• Theorem 1.1: see Theorem \ref{['thm7.A']}
• Definition 3.1: cf. Groenvelt G-2009, sect. 2.1
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Theorem 3.4
• proof
• Definition 4.1
• Definition 4.2