Infinite-dimensional $q$-Jacobi Markov processes
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arXiv
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| Format: | Preprint |
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2025
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| author | Olshanski, Grigori |
| author_facet | Olshanski, Grigori |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_20813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite-dimensional $q$-Jacobi Markov processes Olshanski, Grigori Probability Classical Analysis and ODEs Combinatorics 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. |
| title | Infinite-dimensional $q$-Jacobi Markov processes |
| topic | Probability Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2502.20813 |