Realisation of a class of Markov processes through unitary evolutions in Fock space. (English) Zbl 0743.60061

Séminaire de probabilités, Lect. Notes Math. 1485, 31-36 (1991).
[For the entire collection see Zbl 0733.00018.]
This paper consists of two parts. The first one presents an efficient matrix notation for quantum stochastic calculus on multiple Fock spaces, due to V. P. Belavkin with slight modifications, which has the advantage over the widely used Evans-Hudson notation of having an “Itô table” given by ordinary matrix multiplication (but the involution is different from ordinary matrix adjoint). The second part gives a new example of construction of classical Markov processes as Evans-Hudson non commutative flows: it is assumed that a group \(G\) with Haar measure \(dg\) operates on the state space \(E\), and the generator of the Markov process is of the form \[ Lf(x)=\int_ G(f(gx)-f(x))| \lambda(x,g)|^ 2dg. \]


60H99 Stochastic analysis
81S25 Quantum stochastic calculus
60J99 Markov processes
Full Text: Numdam EuDML