Dilation of Markovian cocycles on a von Neumann algebra

Debashish Goswami, J. Martin Lindsay, Kalyan B. Sinha and Stephen J. Wills

Vol. 211 (2003), No. 2, 221–247

DOI: 10.2140/pjm.2003.211.221

Abstract

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.

Milestones

Received: 15 December 1999

Revised: 10 January 2001

Published: 1 October 2003