Characterization of Dunkl-transform Markov kernels

Characterize precisely the probability measures μ on R^N for which there exists a Markov kernel P related to the Dunkl transform and satisfying μ = P(0,·).

Background

A Markov kernel related to the Dunkl transform is constrained by a multiplicative relation between the Dunkl transform of P(x,·) and the Dunkl kernel. The measure P(0,·) determines the transform multiplier for the kernel.

Although many examples are known, including all pseudo-radial probability measures and the Dunkl Gaussian measures, the paper explicitly states that a complete characterization of admissible probability measures is not known.

References

We notice that it is an open problem, for which probability measures μ∈M1( RN) precisely there exists a Markov kernel P on RN with μ=P(0,.)

— Dunkl theory, convolution algebras, and related Markov processes  (2609.26394 - Rösler et al., 22 Sep 2026) in Section “Markov processes related with integral transforms”, example following Definition of kernels related with an abstract integral transform