Uwe Franz

Simulation of genetic networks modelled by piecewise deterministic Markov processes

The authors propose piecewise deterministic Markov processes as an alternative approach to model gene regulatory networks. A hybrid simulation algorithm is presented and discussed, and several standard regulatory modules are analysed by numerical means. It is shown that despite of the partial simplification of the mesoscopic nature of regulatory networks such processes are suitable to reveal the intrinsic noise effects because of the low copy numbers of genes.

Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

Quantum Markov semigroups (QMS), i.e. strongly continuous semigroups of unital completely positive maps, on compact quantum groups are studied. We show that translation invariant QMSs on the universal or reduced C⁎-algebra of a compact quantum group are in one-to-one correspondence with Levy processes on its ⁎-Hopf algebra. We use the theory of Levy processes on involutive bialgebras to characterize symmetry properties of the associated QMS. It turns out that the QMS is self-adjoint (resp. KMS-symmetric) if and only if the generating functional of the Levy process is invariant under the antipode (resp. the unitary antipode). Furthermore, we study Levy processes whose marginal states are invariant under the adjoint action. Finally, some related aspects of the potential theory as Dirichlet form, derivation and spectral triple are investigated. We discuss how the above results apply to compact groups, group C⁎-algebras of discrete groups, free orthogonal quantum groups On+ and twisted SUq(2) quantum group.

RENORMALIZED POWERS OF QUANTUM WHITE NOISE

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by , where , bs are the Hida white noise densities. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing . This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise.

On idempotent states on quantum groups

Abstract Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed.

Quantum Potential Theory

Potential Theory in Classical Probability.- to Random Walks on Noncommutative Spaces.- Interactions between Quantum Probability and Operator Space Theory.- Dirichlet Forms on Noncommutative Spaces.- Applications of Quantum Stochastic Processes in Quantum Optics.- Quantum Walks.

Stochastic Processes and Operator Calculus on Quantum Groups

Preface. 1. Introduction. 2. Preliminaries on Lie groups. 3. Hopf algebras, quantum groups and braided spaces. 4. Stochastic Processes on quantum groups. 5. Markov Structure of quantum Levy Processes. 6. Diffusions on braided spaces. 7. Evolution equations and Levy processes on quantum groups. 8. Gauss Laws in the sense of Bernstein on quantum groups. 9. Phase retrieval for probability distributions on quantum groups. 10. Limit theorems on quantum groups. Bibliography. Index.

SECOND QUANTIZED AUTOMORPHISMS OF THE RENORMALIZED SQUARE OF WHITE NOISE (RSWN) ALGEBRA

We determine the structure of the *-endomorphisms of the RSWN algebra, induced by linear maps in the 1-particle Hilbert algebra, introduce the RSWN analogue of the free evolutions and find the explicit form of the KMS states associated with some of them.

MONOTONE INDEPENDENCE IS ASSOCIATIVE

It is shown that the notion of monotone independence introduced by Muraki and Lu leads to an associative product of quantum probability spaces. This product is then used to define monotone Levy processes on dual semigroups.

Idempotent states on compact quantum groups and their classification on

Unlike for locally compact groups, idempotent states on locally compact quantum groups do not necessarily arise as Haar states of compact quantum subgroups. We give a simple characterisation of those idempotent states on compact quantum groups which do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups U_q(2), SU_q(2), and SO_q(3) (q in (-1,0) \cup (0,1]) arise in this manner and list the idempotent states on the compact quantum semigroups U_0(2), SU_0(2), and SO_0(3). In the Appendix we provide a short new proof of coamenability of the deformations of classical compact Lie groups based on their representation theory.

\mathrm{U}_q(2)

Abstract We show that idempotent states on finite quantum groups correspond to pre-subgroups in the sense of Baaj, Blanchard, and Skandalis. It follows that the lattices formed by the idempotent states on a finite quantum group and by its coidalgebras are isomorphic. We show, furthermore, that these lattices are also isomorphic for compact quantum groups, if one restricts to expected coidalgebras. To cite this article: U. Franz, A. Skalski, C. R. Acad. Sci. Paris, Ser. I 347 (2009).