Michael Schürmann

White noise on bialgebras

Basic concepts and first results.- Symmetric white noise on Bose Fock space.- Symmetrization.- White noise on bose fock space.- Quadratic components of conditionally positive linear functionals.- Limit theorems.

Quantum independent increment processes on superalgebras

On introduit la notion de processus quantique independant a increment stationnaire sur des superalgebres et on demontre un theoreme de reconstruction qui etablit une correspondance un-a-un entre ces processus et leurs generateurs infinitesimaux

Non-commutative notions of stochastic independence

We present, starting from a set of canonical axioms, a complete classification of the notions of non-commutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schurmann and is based on a fundamental paper by R. Speicher.

Quantum q-white noise and a q-central limit theorem

We establish a connection between the Azema martingales and certain quantum stochastic processes with increments satisfyingq-commutation relations. This leads to a theory ofq-white noise onq-*-bialgebras and to a generalization of the Fock space representation theorem for white noise on *-bialgebras. In particular, quantum Azema noise,q-interpolations between Fermion and Boson quantum Brownian motion and unitary evolutions withq-independent multiplicative increments are studied. It follows from our results that the Azema martingales and theq-interpolations are central limits of sums ofq-independent, identically distributed quantum random variables.

Noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations

SummaryThe notion of a unitary noncommutative stochastic process with independent and stationary increments is introduced, and it is proved that such a process, under a continuity assumption, can be embedded into the solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy [8].

QUANTUM STOCHASTIC CALCULUS ON BOOLEAN FOCK SPACE

In this paper we establish a theory of stochastic integration with respect to the basic field operator processes in the Boolean case. This leads to a Boolean version of quantum Itos product formula and has applications to the theory of dilations of quantum dynamical semigroups.

Infinitesimal Generators on the Quantum Group SUq(2)

Quantum Levy processes on a quantum group are, like classical Levy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SUq(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Levy–Khintchine formula.

A class of representations of involutive bialgebras

A class of representations on Fock space is associated to a representation of the *-algebra structure of a cocommutative graded bialgebra with an involution. We prove that the Gelfand–Naimark–Segal (GNS) representation given by the convolution exponential of a conditionally positive linear functional can be embedded into a representation of this class. Our theory generalizes a well-known construction for infinitely divisible positive definite functions on a group. Applying our general result, we obtain a complete characterization of the GNS representations given by infinitely divisible states on involutive Lie superalgebras.

OPERATOR PROCESSES MAJORIZING THEIR QUADRATIC VARIATION

We give a full classification of convolution semigroups of completely positive mappings on Hopf algebras. Using the theory of noncommutative Levy processes, we prove that these convolution semigroups are solutions of Hudson–Parthasarathy quantum stochastic differential equations. The generating process satisfies a positivity condition on the kernel of the counit which is stronger than complete positivity. It majorizes its bracket process which is the noncommutative process given by the quadratic variation. Our work generalizes and improves parts of the theory of M. Fannes and J. Quaegebeur on infinitely divisible completely positive mappings on groups. It is shown that Azema martingales in the sense of M. Emery arise as components of convolution semigroups on the q-version of the noncommutative polynomial algebra.

Realization of Unitary q-White Noise on Fock Space

We consider families (U t ) t ≥0 of unitary operators on C d ⊗ H, d ∊ ℕ, ℍ a Hilbert space, which have the property that the multiplicative increments for disjoint intervals satisfy certain q-commutation relations with q a complex number of modulus 1. Under additional assumptions on (U t ) t ≥0 which are motivated by classical stochastic processes with independent, stationary increments taking values in the group of unitary d × d-matrices, we give a realization of (U t ) t ≥0 as the solution of a linear quantum stochastic differential equation on Boson Fock space.