## Diffusions with polynomial eigenvectors via finite subgroups of $$O(3)$$.(English. French summary)Zbl 1369.35038

Summary: We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction relies on the finite subgroups of $$O(3)$$ and their invariant polynomials.

### MSC:

 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 60D05 Geometric probability and stochastic geometry 58J65 Diffusion processes and stochastic analysis on manifolds
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### References:

 [1] Bakry (D.), Gentil (I.), and Ledoux (M.).— Analysis and Geometry of Markov Diffusion Operators, Grund. Math. Wiss., vol. 348, Springer, Berlin (2013). [2] Bakry (D.) and Mazet (O.).— Characterization of Markov semigroups on R associated to some families of orthogonal polynomials, Séminaire de Probabilités XXXVII, Lecture Notes in Math., vol. 1832, Springer, Berlin, p. 60-80 (2003). MR MR2053041 [3] Bakry (D.), Orevkov (S.), and Zani (M.).— Orthogonal polynomials and diffusions operators. [4] Meyer (B.).— On the symmetries of spherical harmonics, Canadian J. Math. 6, p. 135-157 (1954). · Zbl 0055.06301 [5] Smith (L.).— Polynomial invariants of finite groups, Research Notes in Mathematics, vol. 6, A K Peters, Ltd., Wellesley, MA (1995). [6] Smith (L.).— Polynomial invariants of finite groups. A survey of recent developments, Bull. Amer. Math. Soc. (N.S.) 34, no. 3, p. 211-250 (1997). · Zbl 0904.13004 [7] Stanley (R. P.).— Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1, no. 3, p. 475-511 (1979). · Zbl 0497.20002
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