Peer-Reviewed Journal Details
Mandatory Fields
Goswami, Debashish; Lindsay, J. Martin; Sinha, Kalyan B.; Wills, Stephen J. ;
Pacific Journal of Mathematics
Dilation of Markovian cocycles on a von Neumann algebra
Optional Fields
Quantum stochastic; completely positive; stochastic flows; quantum dynamical semigroup; locality; dilation; Markovian cocycle
We consider normal Markovian cocycles on a von Neumann algebra which are adapted to a Fock filtration. Every such cocycle k which is Markov-regular and consists of completely positive contractions is realised as a conditioned *-homomorphic cocycle. This amounts to a stochastic generalisation of a recent dilation result for norm-continuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coefficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of cocycles. Number/exchange-free dilatability is seen to be related to locality in the case where the cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markov-regular contraction cocycles on a Hilbert space are also described. The paper ends with a discussion of connections with measure-valued diffusion.
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