ar X iv : m at h - ph / 0 51 20 39 v 1 1 2 D ec 2 00 5 ON STOCHASTIC GENERATORS OF COMPLETELY POSITIVE

A characterisation of the generators of quantum stochastic cocy-cles of completely positive (CP) maps is given in terms of the complete dissipa-tivity (CD) of its form-generator. The pseudo-Hilbert dilation of the stochastic form-generator and the pre-Hilbert dilation of the corresponding dissipator is found. The general form of the linear continuous structural maps for the algebra of all bounded operators is derived and the quantum stochastic flow for the corresponding cocycle is outlined. It is proved that any w*-analytical bounded CD form-generator give rise to a quantum stochastic CP cocycle over a von Neumann algebra. The quantum filtering theory [1] provides examples for a new type of irreversible quantum dynamics, described by one-parameter cocycles: φ = (φ t) t>0 of completely positive stochastic maps φ t (ω) : B → B of an operator algebra B ⊆ B (h). The cocycle condition φ s (ω) • φ r (ω s) = φ r+s (ω) means the stationarity, with respect to the shift ω s = {ω (t + s)} of a given sto-chastic process ω = {ω (t)}. Such maps are in general unbounded, but normalized, φ t (I) = M t to an operator-valued martingale M t = ǫ t [M s ] ≥ 0 with M 0 = 1, or a positive submartingale: M t ≥ ǫ t [M s ], for all s > t, where ǫ t is the conditional expectation with respect to the history up to time t. In the most general case, the stochastically differentiable family φ with respect to a quantum stationary process, with independent increments A s (t) = A (t + s) − A (s) generated by a finite dimensional Itô algebra is described by the quantum stochastic equation dφ t (X) = φ t • α µ ν (X) dA ν µ := µ,ν φ t (α µ ν (X)) dA ν µ , X ∈ B (1) with the initial condition φ 0 (X) = X, for all X ∈ B. Here A ν µ (t) with µ ∈ d} are the standard time A + − (t) = tI, annihilation A m − (t), creation A + n (t) and exchange A m n (t) operator integrators with m, n ∈ {1, ..., d}. The infinitesimal increments dA µ ν (t) = A tµ s (dt) are formally defined