Abstract
A class of quantum stochastic flows on*-algebras is introduced which includes both classical flows on Riemannian manifolds and flows induced by Lie group actions onC *-algebras. Criteria are established to determine those flows which are unitarily equivalent to ones driven by classical Brownian motion. It is shown that taking complex combinations of the driving coefficients of such flows gives rise to flows which are not of Evans-Hudson type (i.e., all driving coefficients do not preserve the relevant algebra).
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References
Applebaum, D. (1988). Quantum stochastic parallel transport processes on noncommutative vector bundles.Springer LNM 1303, 20–37.
Applebaum, D. (1988). Stochastic evolution of Yang-Mills connections on the noncommutative two torus.Lett. Math. Phys. 16, 93–99.
Applebaum, D. (1990). Quantum diffusions on involutive algebras.Quantum Probability and Applications V. Springer LNM 1442, 70–96.
Applebaum, D. (1991). Unitary evolutions and horizontal lifts in quantum stochastic calculus.Commun. Math. Phys. 140, 63–80.
Applebaum, D. (1991). Towards a quantum theory of classical diffusions on Riemannian manifolds.World Scientific, Quantum Probability and Related Topics, pp. 93–111.
Blyth, T. S. (1977).Module Theory, Clarendon Press.
Bratteli, O., Elliott, G. A., Goodman, F. M., and Jorgensen, P. E. T. (1989). Smooth Lie group actions on non-commutative tori.Nonlinearity 2, 271–287.
Bratteli, O., and Robinson, D. W. (1981).Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag.
Connes, A., and Rieffel, M. (1987). Yang Mills for noncommutative two-tori.AMS Contemporary Mathematics 62, 237–267.
Cycon, H. L., Froese, R. G., Kirsch, W., and Simon, B. (1987).Schrödinger Operators, Springer-Verlag.
Evans, M. E. (1989). Existence of quantum diffusions.Prob. Th. Rel. Fields 81, 473–483.
Evans, M. E., and Hudson, R. L. (1988). Multidimensional quantum diffusions.Springer LNM 1303, 68–89.
Fagnola, F. (1990). On quantum stochastic differential equations with unbounded coefficients.Prob. Th. Rel. Fields 86, 501–517.
Hudson, R. L. (1988). Algebraic theory of quantum diffusions.Springer LNM 1325, 113–125.
Hudson, R. L., and Parthasarathy, K. R. (1984). Construction of quantum diffusions.Springer LNM 1055, 173–199.
Hudson, R. L., and Parthasarathy, K. R. (1984). Quantum Ito's formula and stochastic evolution.Commun. Math. Phys. 93, 301–323.
Hudson, R. L., and Robinson, P. (1988). Quantum diffusions and the non-commutative torus.Lett. Math. Phys. 15, 47–53.
Hudson, R. L., and Shepperson, P. (1990). Stochastic dilations of quantum dynamical semigroups using one-dimensional quantum stochastic calculus.Quantum Probability and Applications V, Springer LNM 1442, 216–219.
Lassner, G. (1972). Topological algebras of operators.Rep. Math. Phys. 3, 279–293.
Meyer, P. A. (1989). Chaînes de Markov Finies et Representations Chaotique, Strasbourg (preprint).
Parthasarathy, K. R. (1986). Quantum stochastic calculus,Springer LNM 1203, 177–196.
Parthasarathy, K. R., and Sinha, K. B. (1990). Markov chains as Evans-Hudson diffusions in Fock space.Seminaire de Probabilités XXIV, Springer LNM 1426, 362–369.
Prugovečki, E. (1991). Geometro-stochastic locality in quantum spacetime and quantum diffusions.Found. Phys. 21, 93–124.
Rieffel, M. A. (1981).C *-algebras associated with irrational rotations.Pacific J. Math. 93, 415–429.
Robinson, P. (1990). Quantum diffusions on the rotation algebras and the quantum Hall effect.Quantum Probability and Applications V, Springer LNM 1442, 326–333.
Vincent-Smith, G. F. (1991). Unitary quantum stochastic evolutions.Proc. London Math. Soc. 63, 1–25.
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Applebaum, D. On a class of stochastic flows driven by quantum Brownian motion. J Theor Probab 6, 17–32 (1993). https://doi.org/10.1007/BF01046766
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DOI: https://doi.org/10.1007/BF01046766