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

• Published in: Stochastic processes and their applications 1998-01, Vol.73 (1), p.69-86
• Main Authors: Barchielli, A., Paganoni, A.M., Zucca, F.
• Format: Article
• Language: English
• Subjects: 47D06; 58D25; 60H10; 60H10 58D25 47D06 81P15; 81P15; A posteriori states; Classical and quantum physics: mechanics and fields; Evolution systems; Exact sciences and technology; Foundations, theory of measurement, miscellaneous theories (including aharonov-bohm effect, bell inequalities, berry's phase); Global analysis, analysis on manifolds; Instruments; Master equations; Mathematical analysis; Mathematics; Operator theory; Physics; Probability and statistics; Probability theory and stochastic processes; Quantum mechanics; Sciences and techniques of general use; Semigroups of probability operators; Stochastic analysis; Stochastic differential equations in Hilbert spaces; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(97)00093-8

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.