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Classification of diffusion semigroups on \(\mathbb{R}\) associated with a family of orthogonal polynomials. (Classification des semi-groupes de diffusion sur \(\mathbb{R}\) associés à une famille de polynômes orthogonaux.) (French) Zbl 0883.60072

Azéma, J. (ed.) et al., Séminaire de probabilités XXXI. Berlin: Springer. Lect. Notes Math. 1655, 40-53 (1997).
Let \((P_t, t\geq 0)\) be a strongly continuous Markovian semigroup on \(L^2(I,\mu)\), where \(I\) is an interval of \(\mathbb{R}\) and \(\mu\) a probability measure with finite exponential moment. Assume that the canonical sequence of orthogonal polynomials in \(L^2(I,\mu)\) is the spectral decomposition of \((P_t)\) and that \((P_t)\) is a diffusion semigroup in the sense of Bakry-Emery. It is proved that up to simple transformations these polynomials are either the Hermite polynomials (\(I= \mathbb{R}\)), or the Laguerre polynomials (\(I= \mathbb{R}^*_+\)), or the Jacobi polynomials (\(I= (0, 1)\)). Furthermore, the associated diffusion processes can be obtained through geometrical transformations from the Brownian on the sphere in the Euclidean space.
For the entire collection see [Zbl 0864.00069].

MSC:

60J60 Diffusion processes
60J35 Transition functions, generators and resolvents
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