Dilation of Markovian cocycles on a von Neumann algebra

Goswami, Debashish ; Martin Lindsay, J. ; Sinha, Kalyan B. ; Wills, Stephen J. (2003) Dilation of Markovian cocycles on a von Neumann algebra Pacific Journal of Mathematics, 211 (2). pp. 1-22. ISSN 0030-8730

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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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Deposited On:15 Sep 2011 03:51
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