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Potential kernels for radial Dunkl Laplacians

Abstract : We derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in geometric complex case when the multiplicity k(α) = 1 i.e. for flat complex symmetric spaces. For the invariant Dunkl-Poisson kernel P W (x, y), the estimates are P W (x, y) P R d (x, y) α>0 |x − σ α y| 2k(α) , where the α's are the positive roots of a root system acting in R d , the σ α 's are the corresponding symmetries and P R d is the classical Poisson kernel in R d. Analogous bounds are proven for the Newton kernel when d ≥ 3. The same estimates are derived in the rank one direct product case Z N 2 and conjectured for general W-invariant Dunkl processes. As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.
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Contributor : Piotr Graczyk Connect in order to contact the contributor
Submitted on : Thursday, October 17, 2019 - 3:06:07 PM
Last modification on : Wednesday, November 3, 2021 - 9:18:41 AM
Long-term archiving on: : Saturday, January 18, 2020 - 2:38:31 PM


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  • HAL Id : hal-02318902, version 1



Piotr Graczyk, T Luks, P Sawyer. Potential kernels for radial Dunkl Laplacians. 2019. ⟨hal-02318902⟩



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