The Relation of Spatial and Tensor Product of Arveson Systems --- The Random Set Point of View

We characterise the embedding of the spatial product of two Arveson systems into their tensor product using the random set technique. An important implication is that the spatial tensor product does not depend on the choice of the reference units, i.e. it is an intrinsic construction. There is a continuous range of examples coming from the zero sets of Bessel processes where the two products do not coincide. The lattice of all subsystems of the tensor product is analised in different cases. As a by-product, the Arveson systems coming from Bessel zeros prove to be primitive in the sense of \cite{JMP11a}

Markovian systems of transition expectations

We propose a definition of markovian systems of transition expectation as a generalization of Liebscher’s continuous time version of Accardi’s quantum Markov chains and we show in a reconstruction theorem that the transition expectations may be recovered with the help product systems of Hilbert modules and units for them in the sense of Bhat and Skeide

Segmentation of Time Series: Parameter Dependence of Blake-Zisserman and Mumford-Shah Functionals and the Transition from Discrete to Continuous

The paper deals with variational approaches to the segmentation of time series into smooth pieces, but allowing for sharp breaks. In discrete time, the corresponding functionals are of Blake-Zisserman type. Their natural counterpart in continuous time are the Mumford-Shah functionals. Time series which minimise these functionals are proper estimates or representations of the signals behind recorded data. We focus on consistent behaviour of the functionals and the estimates, as parameters vary or as the sampling rate increases. For each time continuous time series $f\in L^2 (\lbrack 0,1\rbrack)$ we take conditional expectations w.r.t. to $\sigma$-algebras generated by finer and finer partitions of the time domain into intervals, and thereby construct a sequence $(f_n)_{n\in\N}$ of discrete time series. As $n$ increases this amounts to sampling the continuous time series with more and more accuracy. Our main result is consistent behaviour of segmentations w.r.t. to variation of parameters and increasing sampling rate

The Spatial Product of Arveson Systems is Intrinsic

We prove that the spatial product of two spatial Arveson systems is independent of the choice of the reference units. This also answers the same question for the minimal dilation the Powers sum of two spatial CP-semigroups: It is independent up to cocycle conjugacy