Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes

We consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair (SU(N + 1), U(N)); see [I. Pacharoni and J. A. Tirao, Constr. Approx., 25 (2007), pp. 177–192]. After an appropriate conjugation, we obtain a new family of matrix valued orthogonal polynomials where the corresponding block Jacobi matrix is stochastic and has special probabilistic properties. This gives a highly nontrivial example of a nonhomogeneous quasi-birth-and-death process for which we can explicitly compute its “nstep transition probability matrix” and its invariant distribution. The richness of the mathematical structures involved here allows us to give these explicit results for a several parameter family of quasi-birth-and-death processes with an arbitrary (finite) number of phases. Some of these results are plotted to show the effect that choices of the parameter values have on the invariant distribution.Dirección General de Enseñanza SuperiorJunta de Andalucí

Some bivariate stochastic models arising from group representation theory

The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued second-order differential operator). The matrix-valued spherical functions associated to the compact symmetric pair (SU(2)×SU(2),diagSU(2)) will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright–Fisher models involving only mutation effects.Fil: de la Iglesia, Manuel D.. Universidad Nacional Autónoma de México; MéxicoFil: Román, Pablo Manuel. Universidad Nacional de Córdoba; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polyno- mials as its common eigenfunctions leads canonically to a matrix differen- tial operator with the same property. The construction of the correspond- ing family of matrix valued polynomials has been studied in [D1, D2, DV] but the existence of a differential operator having them as common eigen- functions had not been considered This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [GH] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case

Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright-Fisher model

The aim of this paper is to study differential and spectral properties of the infinitesimal operator of two dimensional Markov processes with diffusion and discrete components. The infinitesimal operator is now a second-order differential operator with matrix-valued coefficients, from which we can derive backward and forward equations, a spectral representation of the probability density, study recurrence of the process and the corresponding invariant distribution. All these results are applied to an example coming from group representation theory which can be viewed as a variant of the Wright-Fisher model involving only mutation effects.Comment: 6 figure

QBD processes associated with Jacobi-Koornwinder bivariate polynomials and urn models

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi-Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.Comment: 14 pages, 3 figures. arXiv admin note: text overlap with arXiv:2002.0453

QBD Processes Associated with Jacobi–Koornwinder Bivariate Polynomials and Urn Models

The work of the first author was partially supported by FEDER/Junta de Andalucía under the Research Project A-FQM-246-UGR20; MCIN/AEI 10.13039/501100011033 and FEDER funds by PGC2018-094932-B-I00; and IMAG-María de Maeztu Grant CEX2020-001105-M. The work of the second author was partially supported by PAPIIT-DGAPA-UNAM Grant IN106822 (México) and CONACYT Grant A1-S-16202 (México).We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi–Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.IMAG-María de Maeztu CEX2020-001105-MConsejo Nacional de Ciencia y Tecnología A1-S-16202 CONACYTDirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México IN106822 DGAPA, UNAMFondo de Cooperación Internacional en Ciencia y Tecnología FONCICYTEuropean Regional Development Fund ERDFJunta de Andalucía A-FQM-246-UGR20, PGC2018-094932-B-I0