Bessel convolutions on matrix cones

Abstract

In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras F = R, C or H which interpolate the convolution algebras of radial bounded Borel measures on a matrix space M p,q (F) with p q. Radiality in this context means invariance under the action of the unitary group U p (F) from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize wellknown structures in the rank-one case, namely the Bessel-Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping r → spec(r), the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type B q . The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds U (p, q)/(U p × U q ) over F.

Limit theorems for radial random walks on homogeneous spaces with growing dimensions

“…Again, the proof of Theorem 3.2 is based on a Berry-Esseen inequality on the Euklidean space M p,q ≃ R dpq of [1], while the proof of Theorem 3.3 relies on an explicit formula for the Bessel convolution on Π q due to Rösler [16]. With this convolution, the result follows Similar as for the case q = 1 described in Section 2.…”

“…[14]). Moreover, in case (2), Bessel functions of matrix argument and the associated Bessel convolutions on matrix cones appear (see [6], [8], [16]), and in case (3), Jacobi functions and Jacobi convolutions on [0, ∞[ appear (see the survey [15]). Finally, in the compact case, spheres and projective spaces lead to Jacobi polynomials and Jacobi convolutions on [−1, 1], and the finite examples associated with Hamming schemes or Johnson schemes lead to Krawtchouk and Meixner polynomials respectively.…”

Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

“…e.g., [14]), 2.14 III, [16], Theorem C. See also e.g., [15,3] for applications: T kt/n µ t/n (LT 2) µ t = lim n→∞ (µ(t/n)) n For (matrix cone-) hypergroups we shall prove in analogy to the group case: Theorem 3.2. Let K be a matrix cone hypergroup (investigated in [36,40]) with fixed continuous one parameter group T := (T t ) t≥0 ⊆ Aut(K). In particular, D is again a common core for all continuous convolution semigroups in M 1 * (Γ).…”

“…(Recall that for Lie groups D(G) is just C ∞ c (G).) Recently M. Rösler [36] and M. Voit [40] investigated hypergroup structures on the cone of non-negative definite d × d−matrices with a group like behaviour. In particular, the structure of the automorphism group is well-known, a homomorphic image of GL(R d ).…”

Multiple decomposability of probabilities on contractible locally compact groups

“…For the background the reader is referred to the monograph of W. Bloom and H. Heyer [1]. Recently M. Voit [34] and M. Rösler [24] investigated new classes of hypergroup structures on matrix cones with 'group like' properties. In [5] some basic probabilistic aspects of these hypergroups were investigated.…”