Abstract
We want to describe the triplets (\Omega, (g), \mu) where (g) is the (co)metric associated to some symmetric second order differential operator L defined on the domain \Omega of R^d and such that L is expandable on a basis of orthogonal polynomials of L_2(\mu), and \mu is some admissible measure. Up to affine transformation, we find 11 compact domains in dimension 2, and also give some non--compact cases in this dimension.
Highlights
ABSTRACT. — We study the following problem: describe the triplets (Ω, g, μ) where g = (gij (x)) is themetric associated with the symmetric second order differential operator L(f ) = ρ1 ij ∂i(gij ρ ∂j f ) defined on a domain Ω of Rd (that is L is a diffusion operator with reversible measure μ(dx) = ρ(x)dx) and such that there exists an orthonormal basis of L2(μ) made of polynomials which are at the same time eigenvectors of L, where the polynomials are ranked according to their natural degree
The differential operator L may be replaced by some other generator of a Markov semigroup and the orthogonal polynomial eigenfunctions are Hahn, Krawtchouk, Charlier, Meixner
Under stronger requirements on the sets, we provide a list of the 7 non compact models which solve the problem in dimension 2
Summary
Orthogonal polynomials are a long standing subject of investigation in mathematics. They yield natural Hilbert bases in L2(μ) spaces, where μ is a probability measure on some measurable set Ω in Rd for which polynomials are dense. These polynomial bases are not always the best choice to expand functions or to obtain good approximation schemes This is in particular the case in probability theory, when one is concerned with symmetric diffusion processes as they naturally appear as solutions of stochastic differential equations. We are interested in the description of the situation when the eigenvector expansion coincides with a family of orthogonal polynomials associated with the reversible measure. At least when the set Ω is relatively compact, and when the reversible measure μ has a C1 density with respect to the Lebesgue measure, we may turn the complete description of this situation into a problem of algebraic nature: the operators and the measures can be completely recovered from the boundary of Ω, which is some algebraic surface of degree at most 2d in dimension d.
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