Fock space Markovian cocycles: their representation, generation, and dilation (Q2702411)

That \(k\) is a Markovian cocycle on a unital \(C^*\)-algebra \({\mathcal A}\) is firstly restated in terms of one-parameter semigroup \(\{ {\mathcal P}_t^{[i]} \}\) on \({\mathcal A}\). The characterization of cocycles as weak solutions of quantum stochastic differential equations (QSDEs) is proved when \(k\) is a weakly regular process on \({\mathcal A}\) in \textit{J. M. Lindsay} and \textit{K. R. Parthasarathy} [J. Funct. Anal. 158, No. 2, 521-549 (1998; Zbl 0914.60033)]. In this paper the authors show that \(k\) is a Markovian cocycle on \({\mathcal A}\) if \(k\) is a weakly regular weak solution to the QSDE of similar type, and also give the sufficient conditions for a Markovian cocycle \(k\) on \({\mathcal A}\) to be the unique weakly regular weak solution to the QSDE. Moreover, they characterize the complete positivity of cocycles \(\{ k \}\) when \(k\) is given as a weakly regular weak solution to the QSDE, and prove that the regular Markovian completely positive contraction cocycle \(k\) on \({\mathcal A}\) is a strong solution to the QSDE. A similar characterization can be found in \textit{V. P. Belavkin} [Commun. Math. Phys. 184, No. 3, 533-566 (197; Zbl 0874.60073)]. Lastly given are stochastic dilation theorems showing how completely positive cocycles can be realized in terms of \(*\)-homomorphic cocycles. For other related works, see \textit{R. L. Hudson} and \textit{J. M. Lindsay} [Math. Proc. Camb. Philos. Soc. 102, 363-369 (1987; Zbl 0644.46046)] and \textit{J. M. Lindsay} and \textit{K. B. Sinha} [J. Funct. Anal. 147, No. 2, 400-419 (1997; Zbl 0882.60005)].NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].