Appendix A. Results on semigroups
DOI:
https://doi.org/10.22199/S07160917.1999.0003.00007Abstract
We recall here some results on generation and regularity properties of a strongly continuous semigroup by a second order, possibly degenerated, differential operator.References
[1] L. Accardi: On the quantum Feynman-Kac formula. Rend. Sem. Mat. Fis. Milano, Vol. XLVIII, 135–179 (1978).
[2] L. Accardi: A note on Meyer’s note. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 1–5). Berlin, Heidelberg, New York: Springer 1988.
[3] L. Accardi, C. Cecchini: Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45, (1982), 245–273.
[4] L. Accardi, F. Fagnola, J. Quaegebeur: A representation free quantum stochastic calculus. J. Funct. Anal., 104 (1992), 149–197.
[5] L. Accardi, A. Frigerio, J.T. Lewis: Quantum stochastic processes. Publ. R.I.M.S. Kyoto Univ., 18 (1982), 97–133.
[6] L. Accardi, R. Alicki, A. Frigerio, Y.G. Lu: An invitation to the weak coupling and low density limits. Quantum Probability and Related Topics VI, (1991), 3–62.
[7] L. Accardi, J.L. Journé, J.M. Lindsay: On multidimensional Markovian cocycles. In : Accardi, L., von Waldenfels, W., (eds.) Quantum Probability and Applications IV. Proceedings Rome 1987 (Lect. Notes Math. Vol. 1396, pp. 59-67) Berlin Heidelberg New York, Springer, 1989.
[8] L. Accardi, Y.G. Lu, I.V. Volovich, Quantum Theory and its Stochastic Limit. Springer 2000 (to appear).
[9] L. Accardi, A. Mohari: On the Structure of Classical and Quantum Flows. J. Funct. Anal. 135 (1996), 421–455.
[10] R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications, Lect. Notes Phys. 286, Springer-Verlag, 1987.
[11] D. Applebaum: Towards a Quantum Theory of Classical Di?usions on Riemannian Manifolds. Quantum Probability and Related Topics VI, 93–109, (1991).
[12] W.B. Arveson: Subalgebras of C -algebras, Acta Math., 123 (1969), 141–224.
[13] S. Attal: Extensions of Quantum Stochastic Calculus. Quantum Probability Summer School. Grenoble 1998.
[14] S. Attal, P.-A. Meyer: Interprétation probabiliste et extension des intégrales stochastiques non commutatives. Séminaire de Probabilités, XXVII, 312–327, Lecture Notes in Math., 1557, Springer, Berlin, 1993.
[15] A. Barchielli: Applications of quantum stochastic calculus to quantum optics. Quantum Probability and Related Topics VI, (1991), 111-126.
[16] V.P. Belavkin: A new form and a *-algebraic structure of quantum stochastic integrals in Fock space. Rend. Sem. Mat. Fis. Milano, Vol. LVIII (1988), 177–193.
[17] Ph. Biane: Quelques propriétés du mouvement brownien non-commutatif. Hommage à P. A. Meyer et J. Neveu. Astérisque No. 236 (1996), 73–101.
[18] B.V.R. Bhat, F. Fagnola, K.B. Sinha: On quantum extensions of semigroups of brow- nian motions on an half-line. Russian J. Math. Phys. 4 (1996), 13–28.
[19] B.V.R. Bhat, K.B. Sinha: Examples of unbounded generators leading to nonconservative minimal semigroups. Quantum Probability and Related Topics, IX (1994), 89–104.
[20] B.V.R. Bhat, K.R. Parthasarathy: Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 4, 601–651.
[21] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I. Springer-Verlag, 1979.
[22] A.M. Chebotarev: The theory of conservative dynamical semigroups and applications. MIEM Preprint n.1, Moscow, March 1990.
[23] A.M. Chebotarev: Necessary and su?cient conditions of the conservativism of dynamical semigroups, in: Contemporary Problems , of Mathematics. Newest Achievements 36, VINITI, Moscow (1990), 149–184.
[24] A.M. Chebotarev: Necessary and su?cient conditions for conservativeness of dynamical semigroups, J. Sov. Math., 56 (1991), 2697–2719.
[25] A.M. Chebotarev: Su?cient conditions of the conservativism of a minimal dynamical semigroup. Math. Notes 52 (1993), 1067–1077.
[26] A.M. Chebotarev, F. Fagnola: Su?cient Conditions for Conservativity of Quantum Dynamical Semigroups. J. Funct. Anal. 118 (1993), 131–153.
[27] A.M. Chebotarev, F. Fagnola: On quantum extensions of the Azéma martingale semigroup. Sém. Prob. XXIX (1995), 1–16, Springer LNM 1613.
[28] A.M. Chebotarev, F. Fagnola: Su?cient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups. J. Funct. Anal. 153 (1998), 382–404.
[29] M. Choi: Positive linear maps on C* -algebras, Can. J. Math., XXIV (1972), 520–529.
[30] E. Christensen, D.E. Evans: Cohomology of operator algebras and quantum dynamical semigroups. J. London Math. Soc. 20 (1979), 358–368.
[31] K.L. Chung: Markov Chains with Stationary Transition Probability Springer-Verlag, 1960.
[32] E.B. Davies: Quantum dynamical semigroups and the neutron di?usion equation. Rep. Math. Phys. 11 (1977), 169–188.
[33] S.N. Ethier, T.G. Kurtz: Markov Processes. Characterization and convergence. John Wiley & Sons (1986).
[34] M. Evans: Existence of Quantum Di?usions. Probab. Th. Rel. Fields 81 (1989), 473–483.
[35] D.E. Evans, H. Hanche-Olsen: The generators of positive semi-groups. J. Funct. Anal. 32 (1979), 207–212.
[36] M. Evans, R.L. Hudson: Multidimensional quantum di?usions. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 69–88). Berlin, Heidelberg, New York: Springer 1988.
[37] F. Fagnola: On quantum stochastic di?erential equations with unbounded coe?cients. Probab. Th. Rel. Fields, 86, 501–516 (1990).
[38] F. Fagnola: Pure birth and pure death processes as quantum ?ows in Fock space. Sankhya 53 (1991), 288–297.
[39] F. Fagnola: Unitarity of solutions of quantum stochastic di?erential equations and conservativity of the associated semigroups. Quantum Probability and Related Topics, VII (1992), 139–148.
[40] F. Fagnola: Characterisation of isometric and unitary weakly di?erentiable cocycles in Fock space. Quantum Probability and Related Topics VIII (1993), 143–164.
[41] F. Fagnola: Di?usion processes in Fock space. Quantum Probability and Related Topics IX (1994), 189–214.
[42] F. Fagnola: A simple singular quantum Markov semigroup. To appear in: Anestoc ’98 - Proceedings.
[43] F. Fagnola: Quantum Markov Semigroups and Quantum Markov Flows. Tesi di perfezionamento. Scuola Normale Superiore di Pisa, Pisa 1998.
[44] F. Fagnola, R. Monte: A quantum extension of the semigroup Bessel processes. Mat. Zametki 60 n.5 (1996), p.519–537.
[45] F. Fagnola, R. Rebolledo: The approach to equilibrium of a class of quantum dynamical semigroups. In?nite Dimensional Analysis and Quantum Probability 1, n.4 (1998), 561–572.
[46] F. Fagnola, R. Rebolledo, C. Saavedra: Quantum ?ows associated to master equations in quantum optics. J. Math. Phys. 35 (1994), 1–12.
[47] F. Fagnola, R. Rebolledo, C. Saavedra: Reduction of Noise by squeezed Vacuum. In: R. Rebolledo (ed.), Stochastic Analysis and Mathematical Physics. ANESTOC ’96. World Scienti?c 1998. 61–71.
[48] F. Fagnola, Kalyan B. Sinha: Quantum ?ows with unbounded structure maps and ?nite degrees of freedom. J. London Math. Soc. (2) 48, (1993) p. 537–551, .
[49] W. Feller: On the integro-di?erential equations for purely discontinuous Markov processes. Trans. Am. Math. Soc. 48, 488–575 (1940); Errata 58, 474 (1945).
[50] A. Frigerio: Positive contraction semigroups on B (H) and quantum stochastic differential equations. In: Trends in semigroup theory and applications (Ph. Clement, S. Invernizzi, E. Mitidieri, I.I. Vrabie eds.) Proceedings, Trieste 1987. Marcel Dekker (1989), 175-188.
[51] W. Feller: An Introduction to Probability Theory and Its Applications. Vol. I. John Wiley & Sons, Inc., New York, N.Y., 1950.
[52] A. Frigerio: Some applications of quantum probability to stochastic di?erential equations in Hilbert space. In: Stochastic partial di?erential equations and applications (G. Da Prato and L. Tubaro eds.) Proceedings, Trento 1988. Springer LNM 1390 (1989), 77–90.
[53] J.C. García, R. Quezada: A priori estimates for a class of Quantum Dynamical Semigroups and applications. Cinvestav, Reporte interno n. 235. June 1998.
[54] C.W. Gardiner, P. Zoller: Quantum Noise in Quantum Optics: the Stochastic Schroedinger Equation. http://xxx.sissa.it/list/quant-ph/9702030.
[55] V. Gorini, A. Kossakowski, E.C.G. Sudarshan: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17 (1976), 821–825.
[56] A.S. Holevo: On the structure of covariant dynamical semigroups. J. Funct. Anal. 131 (1995), 255–278.
[57] R.L. Hudson, J.M. Lindsay: On characterizing quantum stochastic evolutions. Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 363–369
[58] R.L. Hudson, K.R. Parthasarathy: Quantum Ito’s formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 301–323.
[59] K. Itô, H.P. McKean, Jr: Di?usion Processes and their Sample Paths, Springer 1965.
[60] J.-L. Journé: Structure des cocycles markoviens sur l’espace de Fock. Probab. Th. Rel. Fields 75 (1987), 291–316.
[61] T. Kato: On the semi-group generated by Kolmogoro?’s di?erential equations, J. Math. Soc. Japan 6 (1954), 1–15.
[62] T. Kato: Perturbation theory for linear operators. Springer-Verlag, 1966.
[63] K. Kraus: General States Changes in Quantum Theory, Ann. Phys., 64 (1970), 311– 335.
[64] G. Lindablad: On the genarators of Quantum DynamicalSemigroups. Commun. Math. Phys. 48 (1976), 119–130.
[65] J.M. Lindsay: Quantum and noncausal stochastic calculus. Probab. Theory Rel. Fields 97 (1993), no. 1-2, 65–80.
[66] H. Maassen: Quantum markov processes on Fock space described by integral kernels. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications II. (Lect. Notes Math., vol. 1136, pp. 361–374) Berlin, Heidelberg, New York: Springer 1985.
[67] P.A. Meyer: A note on shifts and cocycles. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 209–212). Berlin, Heidelberg, New York: Springer 1988.
[68] P.A. Meyer, Quantum Probability for Probabilists, Lect. Notes Math. 1538, Springer- Verlag, 1994.
[69] A. Mohari, K.R. Parthasarathy: On a class of generalizes Evans-Hudson ?ows related to classical markov processes. Quantum Probability and Related Topics, VII (1992), 221–249.
[70] A. Mohari, K.B. Sinha: Stochastic dilation of minimal quantum dynamical semigroup. Proc. Indian Acad. Sci. 102 (1992), 159–173.
[71] R. Monte: Sull’estensione quantistica dei processi di Markov. Università di Palermo. Tesi di dottorato. February 1997.
[72] M.Ohya, D.Petz: Quantum Entropy and its Use, Springer 1995.
[73] P.E.T. Jorgensen: Approximately Reducing Subspaces for Unbounded Linear Operators. J. Funct. Anal. 23 (1976), 392–141.
[74] K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, Vol. 85, 1992.
[75] K.R. Parthasarathy, K.B. Sinha: Markov chains as Evans-Hudson di?usion in Fock space. Sém. Prob. XXIV (1990), 362–369, Springer LNM 1426.
[76] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di?erential Equations, Springer-Verlag, 1975.
[77] D. Petz: Conditional Expectation in Quantum Probability. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Ober- wolfach 1987. (Lect. Notes Math., vol. 1303, pp. 251–260). Berlin, Heidelberg, New York: Springer 1988.
[78] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. I, Functional Analysis, Academic Press, 1975.
[79] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self-Adjointness, Academic Press, 1975.
[80] J.L. Sauvageot: Towards a Quantum Theory of Classical Di?usions on Riemanian Manifolds. Quantum Probability and Related Topics VII, 299–316, (1992).
[81] K.B. Sinha: Quantum Dynamical Semigroups. In: Operator Theory: Advances and Applications, Vol. 70, 161-169, Birkhauser Verlag Basel, 1994.
[82] W.F. Stinespring: Positive functions on C *-algebras, Proc. Am. Math. Soc., 6 (1955), 211–216.
[83] D.W. Stroock, S.R.S. Varadhan: Multidimensional Di?usion Processes. Springer, 1979.
[84] S. Wills: Stochastic Calculus for In?nite Dymensional Noises. Ph. D. Thesis. Notting- ham 1997.
[2] L. Accardi: A note on Meyer’s note. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 1–5). Berlin, Heidelberg, New York: Springer 1988.
[3] L. Accardi, C. Cecchini: Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45, (1982), 245–273.
[4] L. Accardi, F. Fagnola, J. Quaegebeur: A representation free quantum stochastic calculus. J. Funct. Anal., 104 (1992), 149–197.
[5] L. Accardi, A. Frigerio, J.T. Lewis: Quantum stochastic processes. Publ. R.I.M.S. Kyoto Univ., 18 (1982), 97–133.
[6] L. Accardi, R. Alicki, A. Frigerio, Y.G. Lu: An invitation to the weak coupling and low density limits. Quantum Probability and Related Topics VI, (1991), 3–62.
[7] L. Accardi, J.L. Journé, J.M. Lindsay: On multidimensional Markovian cocycles. In : Accardi, L., von Waldenfels, W., (eds.) Quantum Probability and Applications IV. Proceedings Rome 1987 (Lect. Notes Math. Vol. 1396, pp. 59-67) Berlin Heidelberg New York, Springer, 1989.
[8] L. Accardi, Y.G. Lu, I.V. Volovich, Quantum Theory and its Stochastic Limit. Springer 2000 (to appear).
[9] L. Accardi, A. Mohari: On the Structure of Classical and Quantum Flows. J. Funct. Anal. 135 (1996), 421–455.
[10] R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications, Lect. Notes Phys. 286, Springer-Verlag, 1987.
[11] D. Applebaum: Towards a Quantum Theory of Classical Di?usions on Riemannian Manifolds. Quantum Probability and Related Topics VI, 93–109, (1991).
[12] W.B. Arveson: Subalgebras of C -algebras, Acta Math., 123 (1969), 141–224.
[13] S. Attal: Extensions of Quantum Stochastic Calculus. Quantum Probability Summer School. Grenoble 1998.
[14] S. Attal, P.-A. Meyer: Interprétation probabiliste et extension des intégrales stochastiques non commutatives. Séminaire de Probabilités, XXVII, 312–327, Lecture Notes in Math., 1557, Springer, Berlin, 1993.
[15] A. Barchielli: Applications of quantum stochastic calculus to quantum optics. Quantum Probability and Related Topics VI, (1991), 111-126.
[16] V.P. Belavkin: A new form and a *-algebraic structure of quantum stochastic integrals in Fock space. Rend. Sem. Mat. Fis. Milano, Vol. LVIII (1988), 177–193.
[17] Ph. Biane: Quelques propriétés du mouvement brownien non-commutatif. Hommage à P. A. Meyer et J. Neveu. Astérisque No. 236 (1996), 73–101.
[18] B.V.R. Bhat, F. Fagnola, K.B. Sinha: On quantum extensions of semigroups of brow- nian motions on an half-line. Russian J. Math. Phys. 4 (1996), 13–28.
[19] B.V.R. Bhat, K.B. Sinha: Examples of unbounded generators leading to nonconservative minimal semigroups. Quantum Probability and Related Topics, IX (1994), 89–104.
[20] B.V.R. Bhat, K.R. Parthasarathy: Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 4, 601–651.
[21] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I. Springer-Verlag, 1979.
[22] A.M. Chebotarev: The theory of conservative dynamical semigroups and applications. MIEM Preprint n.1, Moscow, March 1990.
[23] A.M. Chebotarev: Necessary and su?cient conditions of the conservativism of dynamical semigroups, in: Contemporary Problems , of Mathematics. Newest Achievements 36, VINITI, Moscow (1990), 149–184.
[24] A.M. Chebotarev: Necessary and su?cient conditions for conservativeness of dynamical semigroups, J. Sov. Math., 56 (1991), 2697–2719.
[25] A.M. Chebotarev: Su?cient conditions of the conservativism of a minimal dynamical semigroup. Math. Notes 52 (1993), 1067–1077.
[26] A.M. Chebotarev, F. Fagnola: Su?cient Conditions for Conservativity of Quantum Dynamical Semigroups. J. Funct. Anal. 118 (1993), 131–153.
[27] A.M. Chebotarev, F. Fagnola: On quantum extensions of the Azéma martingale semigroup. Sém. Prob. XXIX (1995), 1–16, Springer LNM 1613.
[28] A.M. Chebotarev, F. Fagnola: Su?cient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups. J. Funct. Anal. 153 (1998), 382–404.
[29] M. Choi: Positive linear maps on C* -algebras, Can. J. Math., XXIV (1972), 520–529.
[30] E. Christensen, D.E. Evans: Cohomology of operator algebras and quantum dynamical semigroups. J. London Math. Soc. 20 (1979), 358–368.
[31] K.L. Chung: Markov Chains with Stationary Transition Probability Springer-Verlag, 1960.
[32] E.B. Davies: Quantum dynamical semigroups and the neutron di?usion equation. Rep. Math. Phys. 11 (1977), 169–188.
[33] S.N. Ethier, T.G. Kurtz: Markov Processes. Characterization and convergence. John Wiley & Sons (1986).
[34] M. Evans: Existence of Quantum Di?usions. Probab. Th. Rel. Fields 81 (1989), 473–483.
[35] D.E. Evans, H. Hanche-Olsen: The generators of positive semi-groups. J. Funct. Anal. 32 (1979), 207–212.
[36] M. Evans, R.L. Hudson: Multidimensional quantum di?usions. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 69–88). Berlin, Heidelberg, New York: Springer 1988.
[37] F. Fagnola: On quantum stochastic di?erential equations with unbounded coe?cients. Probab. Th. Rel. Fields, 86, 501–516 (1990).
[38] F. Fagnola: Pure birth and pure death processes as quantum ?ows in Fock space. Sankhya 53 (1991), 288–297.
[39] F. Fagnola: Unitarity of solutions of quantum stochastic di?erential equations and conservativity of the associated semigroups. Quantum Probability and Related Topics, VII (1992), 139–148.
[40] F. Fagnola: Characterisation of isometric and unitary weakly di?erentiable cocycles in Fock space. Quantum Probability and Related Topics VIII (1993), 143–164.
[41] F. Fagnola: Di?usion processes in Fock space. Quantum Probability and Related Topics IX (1994), 189–214.
[42] F. Fagnola: A simple singular quantum Markov semigroup. To appear in: Anestoc ’98 - Proceedings.
[43] F. Fagnola: Quantum Markov Semigroups and Quantum Markov Flows. Tesi di perfezionamento. Scuola Normale Superiore di Pisa, Pisa 1998.
[44] F. Fagnola, R. Monte: A quantum extension of the semigroup Bessel processes. Mat. Zametki 60 n.5 (1996), p.519–537.
[45] F. Fagnola, R. Rebolledo: The approach to equilibrium of a class of quantum dynamical semigroups. In?nite Dimensional Analysis and Quantum Probability 1, n.4 (1998), 561–572.
[46] F. Fagnola, R. Rebolledo, C. Saavedra: Quantum ?ows associated to master equations in quantum optics. J. Math. Phys. 35 (1994), 1–12.
[47] F. Fagnola, R. Rebolledo, C. Saavedra: Reduction of Noise by squeezed Vacuum. In: R. Rebolledo (ed.), Stochastic Analysis and Mathematical Physics. ANESTOC ’96. World Scienti?c 1998. 61–71.
[48] F. Fagnola, Kalyan B. Sinha: Quantum ?ows with unbounded structure maps and ?nite degrees of freedom. J. London Math. Soc. (2) 48, (1993) p. 537–551, .
[49] W. Feller: On the integro-di?erential equations for purely discontinuous Markov processes. Trans. Am. Math. Soc. 48, 488–575 (1940); Errata 58, 474 (1945).
[50] A. Frigerio: Positive contraction semigroups on B (H) and quantum stochastic differential equations. In: Trends in semigroup theory and applications (Ph. Clement, S. Invernizzi, E. Mitidieri, I.I. Vrabie eds.) Proceedings, Trieste 1987. Marcel Dekker (1989), 175-188.
[51] W. Feller: An Introduction to Probability Theory and Its Applications. Vol. I. John Wiley & Sons, Inc., New York, N.Y., 1950.
[52] A. Frigerio: Some applications of quantum probability to stochastic di?erential equations in Hilbert space. In: Stochastic partial di?erential equations and applications (G. Da Prato and L. Tubaro eds.) Proceedings, Trento 1988. Springer LNM 1390 (1989), 77–90.
[53] J.C. García, R. Quezada: A priori estimates for a class of Quantum Dynamical Semigroups and applications. Cinvestav, Reporte interno n. 235. June 1998.
[54] C.W. Gardiner, P. Zoller: Quantum Noise in Quantum Optics: the Stochastic Schroedinger Equation. http://xxx.sissa.it/list/quant-ph/9702030.
[55] V. Gorini, A. Kossakowski, E.C.G. Sudarshan: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17 (1976), 821–825.
[56] A.S. Holevo: On the structure of covariant dynamical semigroups. J. Funct. Anal. 131 (1995), 255–278.
[57] R.L. Hudson, J.M. Lindsay: On characterizing quantum stochastic evolutions. Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 363–369
[58] R.L. Hudson, K.R. Parthasarathy: Quantum Ito’s formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 301–323.
[59] K. Itô, H.P. McKean, Jr: Di?usion Processes and their Sample Paths, Springer 1965.
[60] J.-L. Journé: Structure des cocycles markoviens sur l’espace de Fock. Probab. Th. Rel. Fields 75 (1987), 291–316.
[61] T. Kato: On the semi-group generated by Kolmogoro?’s di?erential equations, J. Math. Soc. Japan 6 (1954), 1–15.
[62] T. Kato: Perturbation theory for linear operators. Springer-Verlag, 1966.
[63] K. Kraus: General States Changes in Quantum Theory, Ann. Phys., 64 (1970), 311– 335.
[64] G. Lindablad: On the genarators of Quantum DynamicalSemigroups. Commun. Math. Phys. 48 (1976), 119–130.
[65] J.M. Lindsay: Quantum and noncausal stochastic calculus. Probab. Theory Rel. Fields 97 (1993), no. 1-2, 65–80.
[66] H. Maassen: Quantum markov processes on Fock space described by integral kernels. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications II. (Lect. Notes Math., vol. 1136, pp. 361–374) Berlin, Heidelberg, New York: Springer 1985.
[67] P.A. Meyer: A note on shifts and cocycles. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 209–212). Berlin, Heidelberg, New York: Springer 1988.
[68] P.A. Meyer, Quantum Probability for Probabilists, Lect. Notes Math. 1538, Springer- Verlag, 1994.
[69] A. Mohari, K.R. Parthasarathy: On a class of generalizes Evans-Hudson ?ows related to classical markov processes. Quantum Probability and Related Topics, VII (1992), 221–249.
[70] A. Mohari, K.B. Sinha: Stochastic dilation of minimal quantum dynamical semigroup. Proc. Indian Acad. Sci. 102 (1992), 159–173.
[71] R. Monte: Sull’estensione quantistica dei processi di Markov. Università di Palermo. Tesi di dottorato. February 1997.
[72] M.Ohya, D.Petz: Quantum Entropy and its Use, Springer 1995.
[73] P.E.T. Jorgensen: Approximately Reducing Subspaces for Unbounded Linear Operators. J. Funct. Anal. 23 (1976), 392–141.
[74] K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, Vol. 85, 1992.
[75] K.R. Parthasarathy, K.B. Sinha: Markov chains as Evans-Hudson di?usion in Fock space. Sém. Prob. XXIV (1990), 362–369, Springer LNM 1426.
[76] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di?erential Equations, Springer-Verlag, 1975.
[77] D. Petz: Conditional Expectation in Quantum Probability. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications III. Proceedings, Ober- wolfach 1987. (Lect. Notes Math., vol. 1303, pp. 251–260). Berlin, Heidelberg, New York: Springer 1988.
[78] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. I, Functional Analysis, Academic Press, 1975.
[79] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self-Adjointness, Academic Press, 1975.
[80] J.L. Sauvageot: Towards a Quantum Theory of Classical Di?usions on Riemanian Manifolds. Quantum Probability and Related Topics VII, 299–316, (1992).
[81] K.B. Sinha: Quantum Dynamical Semigroups. In: Operator Theory: Advances and Applications, Vol. 70, 161-169, Birkhauser Verlag Basel, 1994.
[82] W.F. Stinespring: Positive functions on C *-algebras, Proc. Am. Math. Soc., 6 (1955), 211–216.
[83] D.W. Stroock, S.R.S. Varadhan: Multidimensional Di?usion Processes. Springer, 1979.
[84] S. Wills: Stochastic Calculus for In?nite Dymensional Noises. Ph. D. Thesis. Notting- ham 1997.
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2018-04-04
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[1]
F. Fagnola, “Appendix A. Results on semigroups”, Proyecciones (Antofagasta, On line), vol. 18, no. 3, pp. 135-144, Apr. 2018.
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