Markov chains as Evans-Hudson diffusions in Fock space. (English) Zbl 0699.60065

Séminaire de probabilités XXIV 1988/89, Lect. Notes Math. 1426, 362-369 (1990).
[For the entire collection see Zbl 0695.00024.]
Let \({\mathcal A}_ 0\) be a von-Neumann algebra of operators in a Hilbert space and let \(\tau^ i_ j: {\mathcal A}_ 0\to {\mathcal A}_ 0\) be bounded linear maps for \(0\leq i,j\leq n\) satisfying \[ \tau^ i_ j(XY)=\tau^ i_ j(X)Y+X\tau^ i_ j(Y)+\sum^{n}_{k=1}\tau^ i_ k(X)t^ k_ j(Y), \]
\[ \tau^ i_ j(X)^*=\tau^ j_ i(X^*),\quad \tau^ i_ j(1)=0,\quad \| \tau^ i_ j(X)\| \leq M \| X\| \] for all \(0\leq i,j\leq n\), \(X,Y\in {\mathcal A}_ 0\), M being a positive constant. It is then known from the work of Evans and Hudson that there exists a family \(\{j_ t\), \(t\geq 0\}\) of * unital homomorphisms from \({\mathcal A}_ 0\) into \({\mathcal A}_ 0\otimes {\mathcal B}(\Gamma (L^ 2({\mathbb{R}}_+)\otimes C^ n)\) satisfying a quantum stochastic differential equation \[ dj_ t(X)=\sum_{i,k}j_ t(\tau^ i_ k(X))d\Lambda^ k_ i(t) \] where \(\{\Lambda^ k_ i\), \(0\leq i,k\leq n\), \(i+k>0\}\) are the basic noise processes in the boson Fock space \(\Gamma (L^ 2({\mathbb{R}}_+)\otimes C^ n)\) and \(d\Lambda^ 0_ 0(t)=dt.\)
It is shown here that when \({\mathcal A}_ 0\) is abelian the family \(\{j_ t(X)\), \(t\geq 0\), \(X\in {\mathcal A}_ 0\}\) of operators is also abelian. This indicates the possibility of realising classical Markov flows by a suitable choice of \(\{\tau^ i_ k\}\) on the abelian algebra of functions on a Borel space and the construction of \(\{j_ t\}\). To lend credence to such a belief a class of Markov chains with a finite or countable state space is constructed on these lines by a suitable combination of an imprimitivity system in the sense of Mackey and a unitary operator-valued process obeying a quantum stochastic differential equation with respect to \(\{\Lambda^ i_ j\}\).
Reviewer: K.R.Parthasarathy


60J27 Continuous-time Markov processes on discrete state spaces
60H99 Stochastic analysis
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras


Zbl 0695.00024
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