QUANTUM MARKOV AND NONLINEAR SEMIGROUPS

Quantum Markov Semigroups (QMS)
in continuous time were studied by Davies in the beginning of 70s as the simplest models of QSP, but real progress started after the discovery of Lindblad form for their bounded generators. I extended the Lindblad's theorem to the nonlinear completely positive semigroups and constructed in [1] minimal solutions to quantum Master equations with relatively bounded generators describing quantum branching processes by nonlinear analytical generators of this form. Later Chebotarev and Fagnola found sufficient conservativity conditions for some Markov semigroups with unbounded generators using the solutions in this form. The dilation of the unbounded Lindblad form-generators as conditionally positive-definite sesquilinear forms was constructed in [2], and for the dilation of the conditionally positive-definite stochastic germs corresponding to Quantum Master Equations see [3].
Quantum Stochastic Cocycles(QSC)
are generalizations of QMS to Quantum Markov Flows (QMF), the shift-traslated semigroups of completely positive quantum stochastic maps. Based on [2,3] the Lindblad-like germs for these flows were discovered and classified in [4,5]. The minimal solution for the corresponding linear evolution equations was constructed for finite and infinite-dimensional Ito algebras and even for unbounded generators corresponding to contractive and filtering folws in [6,7]. Lindsay and Parthasarathy put this into a more traditional framework of QS calculus based on simple (i.e. vacuum, or HP) Ito algebra, and recently Lindsay with Wills extended my results for contractive CP flows to arbitrary initial algebra (resticting them to simple Ito algebra and bounded generators) using another series method for construction the solutions to such equations with bounded generators. My dilation theorem for quantum stochastic conditionally-positive generators and cocycles with respect to an arbitrary (infinite-dimensional) Ito algebra extends the quantum stochastic dilations of quantum Markov semigroups to Hudson-Evens flows. For application of this theory to contractive and filtering quantum stochastic CP flows see QEE and QMF pages.
Quantum Kinetic Equations (QKE)
generate nonlinear quantum evolutions. The discovered structure of linear quantum Master equations can be applied for canonical derivation of nonlinear QKE asymptotically describing quantum Markov diffusion-reaction processes. Such equation were derived for purely branching quantum Markov particles in [8], and for weakly interacting particles in [9]. Even earlier I proved in the series of papers {10-12] that under condition of weak innteraction, such models can be reduced to the nonlinear single-particle dynamics of Bolzmann and Vlasov type by use of the quantum law of large numbers. Currently I am interested in quantum stochastic extension of these results, dilations of the kinetic equations, and the quantum large deviations. Also other important results concerned with the corresponding central limit theorems and large deviations, can be obtained in this direction.
My 10 Relevant Publications:
  1. V. P. Belavkin: Multiquantum Systems and Point Processes 1. Generating Functionals and Non-Linear Semigroups. Reports on Math Phys 28 (1) 57--90 (1989).
  2. V. P. Belavkin: A Pseudoeucledean Representation of Conditionally Positive Maps. Mathematical Notes 49 (6) 135--137 (1991).
  3. V. P. Belavkin: Positive Definite Germs of Quantum Stochastic Processes. C. R. Acad. Sci. Paris 322 385-390 (1996). math.PR/0512289, PDF.
  4. V. P. Belavkin: On Stochastic Generators of Completely Positive Cocycles. Russ Journ of Math Phys 3 (4) 523--528 (1995). math-ph/0512039, PDF.
  5. V. P. Belavkin: On the General Form of Quantum Stochastic Evolution Equation. In Stochastic Analysis and Applications 91--106, World Scientific, Singapore, 1996. math.PR/0512510, PDF.
  6. V. P. Belavkin: Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation. Commun. Math. Phys. 184 533-566 (1997). math-ph/0512042, PDF.
  7. V. P. Belavkin: Quantum Stochastic Semigroups and Their Generators. In Jrreversibility and Causality 82-109. Springer Verlag, Lecture Notes in Physics, Berlin, 1998. math.PR/0512360, PDF.
  8. V. P. Belavkin: Non-Commutative Point Processes and Quantum Kinetic Equations. Proc of 1st World Congress of the Bernoulli Society II 473--478, Science Press, VNU, Utrecht, 1988.
  9. V. P. Belavkin: Quantum Branching Processes and Nonlinear Dynamics of Multiquantum Systems. Soviet Math Dokl 301 (6) 1348--1352 (1988).
  10. V. P. Belavkin: Markovian Dynamics of Quantum Systems and Quantum Kinetic Equations. In Proc of U S S R School on Mathematical Models in Statistical Physics 3--12, University of Tiumen', 1982.
  11. V. P. Belavkin, V. P. Maslov and S. Tariverdiev: Derivation of Bolzman Equation from Kolmogorov--Feller Equation. Theoret Math Phys 49 (3) 298--306 (1981).
  12. V. P. Belavkin and V. P. Maslov: Uniformization Method in the Theory of Nonlinear Hamiltonian Vlasov and Hartree Type Systems. Theoret Math Physics 33 (1) 17--31 (1977). math-ph/0512051, PDF.