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On a Relation Between Classical and Free Infinitely Divisible Transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

349--367

We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.

Martingale characterizations of stochastic processes on compact groups

Voit M.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

389--405

By a classical result of P. Lévy, the Brownian motion (Bt)t≥0 on R may be characterized as a continuous process on R such that (Bt)t≥0 and (B2t - t)t≥0 are martingales. Generalizations of this result are usually obtained in the setting of the so-called martingale problem. This paper contains a variant of the martingale problem for stochastic processes on locally compact groups with independent stationary increments that is based on irreducible unitary representations. In particular, for Gaussian processes on compact Lie groups, analogues of the Lévy-characterization above are obtained. It turns out that for certain compact Lie groups even the continuity assumption in this characterization can be dropped.

Decomposition of Convolution Semigroups on Groups and the 0-1 Law

Byczkowska H., Byczkowski T.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 1

97--104

Let (X(t))t>0, be a stochastically continuous symmetric Lévy process with values in a complete separable group G. We denote by (μt)t>0 the semigroup of one-dimensional distributions of X(t). Suppose that H is a Borel subgroup of G such that μt (H) > 0 for all t > 0. We obtain a decomposition of the generator of the process X ( t ) into a bounded part concentrated on Hc and the generator of a semigroup concentrated on H. This yields the 0-1 law for such processes. We also examine the differentiation of transition probability of the induced Markov process π (X (t)) on the homogeneous space G/H.

A Lévy Khintchine type representation of convolution semigroups : on commutative hypergroups

Rentzsch C.

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

185--198

We first study Lévy measures, Poisson and Gaussian convolution semigroups on commutative hypergroups. Then we present a Lévy-Khintchine type representation of a convolution semigroup (μt)t>0with symmetric Lévy measure λ of the form μt = γt,*e(tγ), t≥0, for some Poisson semigroup (e(tγ)) t>0 and some Gaussian semigroup (γt) t>0.