On stochastic differential equations and semigroups of probability operators in quantum probability

Some “classical” stochastic differential equations have been used in the theory of measurements continuous in time in quantum mechanics and, more generally, in quantum open system theory. In this paper, we introduce and study a class of such equations which allow us to achieve the same level of generality as the one obtained by the approach to continuous measurements based on semigroups of operators. To this aim, we have to study some linear and non-linear stochastic differential equations for processes in Hilbert spaces and in some related Banach spaces. By this stochastic approach we can also obtain new results on the evolution systems which substitute the semigroups of probability operators in the time inhomogeneous case.