On the phenomena of constant curvature in the diffusion-orthogonal\n polynomials

2014/09/18 by Lev Soukhanov, Soukhanov, Lev
Mathematics · #Differential Equations and Boundary Problems #advanced mathematical theories #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.1409.5332

Abstract

We consider the systems of diffusion-orthogonal polynomials, defined in the\nwork [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why\nthese systems with boundary of maximal possible degree should always come from\nthe group, generated by reflections. Our proof works for the dimensions 2 (on\nwhich this phenomena was discovered) and 3, and fails in the dimensions 4\nand higher, leaving the possibility of existence of diffusion-orthogonal\nsystems related to the Einstein metrics.\n The methods of our proof are algebraic / complex analytic in nature and based\nmainly on the consideration of the double covering of \ℂ^d, branched\nin the boundary divisor.\n Author wants to thank Stepan Orevkov, Misha Verbitsky and Dmitry Korb for\nuseful discussions.\n

Related

• Orthogonal polynomials and diffusion operators
• Symmetric diffusions with polynomial eigenvectors
• Rigidity phenomenons for an infinite dimension diffusion operator and\n cases of near equality in the Bakry--Ledoux isoperimetric comparison Theorem
• Modified Curvatures on Manifolds with Boundary and Applications
• Rigidity for Markovian Maximal Couplings of Elliptic Diffusions