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Original Articles

Dilations à la Hudson–Parthasarathy of Markov Semigroups in Classical Probability

Pages 1025-1052 | Accepted 17 Dec 2007, Published online: 28 Aug 2008

REFERENCES

  • Accardi , L. , Lu , Y.G. , and Volovich , I. 2002 . Quantum Theory and Its Stochastic Limit . Springer-Verlag , Berlin .
  • Baccelli , F. , and Brèmaud , P. 1987 . Palm Probabilities and Stationary Queues . Lecture Notes in Statistics, 41 . Springer-Verlag , Berlin .
  • Chebotarev , A.M. 1997 . The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation . Mathematica Notes 61 : 510 – 518 .
  • Chebotarev , A.M. 1998 . Quantum stochastic differential equation is unitary equivalent to a symmetric boundary value problem in Fock space . Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 : 175 – 199 .
  • Ethier , S.N. , and Kurtz , T.G. 1986 . Markov Processes. Characterization and Convergence . John Wiley & Sons, Inc. , New York .
  • Frigerio , A. 1985 . Covariant Markov dilations of quantum dynamical semigroups . Publ. RIMS Kyoto Univ. 21 : 657 – 675 .
  • Frigerio , A. 1985. Construction of stationary quantum Markov processes through quantum stochastic calculus. In: Quantum Probability and Applications II . Lecture Notes in Mathematics, 1136. Springer-Verlag , Berlin, 207–222.
  • Gregoratti , M. 2000 . On the Hamiltonian operator associated to some quantum stochastic differential equations . Infinite Dimensional Analysis, Quantum Probability and Related Topics 3 ( 4 ): 483 – 503 .
  • Gregoratti , M. 2001 . The Hamiltonian operator associated with some quantum stochastic evolutions . Commun. Math. Phys. 222 : 181 – 200 .
  • Gregoratti , M. 2008 . Dilations à la Quantum Probability of Markov evolutions in discrete time. Quaderno di Dipartimento QDD 14, math.PR/0702690v1 . Theory of Probability and Its Applications 53 ( 4 ).
  • Hudson , R.L. , and Parthasarathy , K.R. 1984 . Quantum Itô's formula and stochastic evolutions . Commun. Math. Phys. 93 : 301 – 323 .
  • Karr , A.F. 1991 . Point Processes and Their Statistical Inference . Marcel Dekker , New York .
  • Lewis , J.T. , and Maassen , H. 1984 . Hamiltonian models of classical and quantum stochastic processes . In: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes (Villa Mondragone, 1982) . Lecture Notes in Mathematics, 1055 . Springer , Berlin , 245 – 276 .
  • Maassen , H. 1984 . The construction of continuous dilations by solving quantum stochastic differential equations . Semesterbericht Funktionalanalysis Tübingen Sommersemester 84 : 183 – 204 .
  • Maassen , H. 1985 . Quantum Markov processes on Fock space described by integral kernels . In: Quantum Probability and Applications II . Lecture Notes in Mathematics 1136 . Springer-Verlag , Berlin , 361 – 374 .
  • Meyer , P.A. 1993 . Quantum Probability for Probabilists , Lecture Notes in Mathematics, 1538 . Springer-Verlag , Berlin .
  • Mohary , A. , and Sinha , K.B. 1990 . Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes . Sankhyā 52 ( A Pt.1 ): 43 – 57 .
  • Parthasarathy , K.R. 1992 . An Introduction to Quantum Stochastic Calculus . Birkhäuser , Basel .
  • Rozanov , Y.A. 1977 . Innovation Processes . V.H. Winston & Sons , Washington , DC .
  • Tsirelson , B. 1998 . Within and beyond the reach of Brownian innovation . In: Proceedings of the International Congress of Mathematicians, Vol. III, Berlin, 1998 . Doc. Math. Extra Vol. III , 311 – 320 .

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