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Characterization of Markov semigroups on \(\mathbb{R}\) associated to some families of orthogonal polynomials. (English) Zbl 1060.33014

Azéma, J. (ed.) et al., 37th seminar on probability. Berlin: Springer (ISBN 3-540-20520-9/pbk). Lect. Notes Math. 1832, 60-80 (2003).
A measure \(\mu\), not supported on any \( (-\infty, M] \), is called exponentially integrable if there exists \( \varepsilon >0\) such that \( \int e^{\varepsilon | x| } \mu (dx)< \infty \). Let \((P_n)\) be a family of orthogonal polynomials on \(L^2 (\mu)\). If \((P_n)\) are the eigenfunctions of the transition kernel of a Markov chain with eigenvalues \(c_n\), then \(c_n\) is called a Markov sequence associated to \((P_n)\). For the continuous time case, \(\lambda_n\) is called a Markov generator sequence associated to \((P_n)\), if \(\lambda_n\) and \((P_n)\) are the eigenvalues and eigenfunctions of the generator of a reversible Markov process. In the present paper, under some appropriate conditions, the expression of the Markov sequence \(C_N\) and Markov generator sequence \(\lambda_n\) are obtained. As special interesting cases of the Hermite polynomials, the Laguerre polynomials, the Charlier polynomials and the Meixner polynomials are analyzed, where the reference measures of the last two are supported on the integers.
For the entire collection see [Zbl 1027.00025].

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
47D07 Markov semigroups and applications to diffusion processes
33C90 Applications of hypergeometric functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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