Markov chains approximation of jump-diffusion stochastic master equations
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, p. 924-948

Quantum trajectories are solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also called Belavkin equations or Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump-diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab. 36 (2008) 2332-2353] and [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.

Les trajectoires quantiques sont des solutions d'équations différentielles stochastiques décrivant des phénomènes aléatoires associés aux principes de mesure (quantique) des systèmes quantiques ouverts. Ces équations, également appelées équations de Belavkin ou équations maîtresses stochastiques sont habituellement de deux types: soit diffusif soit de type saut. Dans cet article, nous considérons des modèles plus avancés où des équations de type saut-diffusion apparaissent. Ces équations sont obtenues comme solutions de problèmes de martingales. Ces problèmes de martingales sont obtenus comme limites continus (en temps) à partir de chaînes de Markov classiques décrivant des trajectoires quantiques pour des modèles à temps discret. Les résultats de cet article généralisent ceux de [Ann. Probab. 36 (2008) 2332-2353] et [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. Ici, les techniques probabilistes utilisés sont complétement différentes afin de pouvoir mixer les deux types d'évolutions: diffusives et poissoniennes.

DOI : https://doi.org/10.1214/09-AIHP330
Classification:  60F99,  60G99,  60H10
Keywords: stochastic master equations, quantum trajectory, Jump-diffusion stochastic differential equation, stochastic convergence, Markov generators, martingale problem
@article{AIHPB_2010__46_4_924_0,
     author = {Pellegrini, Cl\'ement},
     title = {Markov chains approximation of jump-diffusion stochastic master equations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {924-948},
     doi = {10.1214/09-AIHP330},
     zbl = {1211.60020},
     mrnumber = {2744878},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_4_924_0}
}
Pellegrini, Clément. Markov chains approximation of jump-diffusion stochastic master equations. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 924-948. doi : 10.1214/09-AIHP330. http://www.numdam.org/item/AIHPB_2010__46_4_924_0/

[1] S. Attal and Y. Pautrat. From repeated to continuous quantum interactions. Ann. Henri Poincaré 7(1) (2006) 59-104. | MR 2205464 | Zbl 1099.81040

[2] A. Barchielli and M. Gregoratti. Quantum Trajectories and Measurements in Continuous Time-the Diffusive Case. Lecture Notes in Physics 782. Springer, Berlin, 2009. | MR 2841028 | Zbl 1182.81001

[3] A. Barchielli and V. P. Belavkin. Measurements continuous in time and a posteriori states in quantum mechanics. J. Phys. A 24(7) (1991) 1495-1514. | MR 1121822

[4] A. Barchielli and A. M. Paganoni. Stochastic differential equations for trace-class operators and quantum continual measurements. In Stochastic Partial Differential Equations and Applications (Trento, 2002). Lecture Notes in Pure and Appl. Math. 227 53-67. Dekker, New York, 2002. | MR 1919502 | Zbl 1004.60058

[5] A. Barchielli, A. M. Paganoni and F. Zucca. On stochastic differential equations and semigroups of probability operators in quantum probability. Stochastic Process. Appl. 73(1) (1998) 69-86. | MR 1603834 | Zbl 0940.60078

[6] A. Barchielli and F. Zucca. On a class of stochastic differential equations used in quantum optics. Rend. Sem. Mat. Fis. Milano 66 (1998) 355-376. | MR 1639788 | Zbl 0920.60040

[7] A. Barchielli and A. S. Holevo. Constructing quantum measurement processes via classical stochastic calculus. Stochastic Process. Appl. 58 (1995) 293-317. | MR 1348380 | Zbl 0829.60050

[8] R. F. Bass. Stochastic differential equations with jumps. Probab. Surv. 1 (2004) 1-19 (electronic). | MR 2095564 | Zbl 1189.60114

[9] V. P. Belavkin. Quantum stochastic calculus and quantum nonlinear filtering. J. Multivariate Anal. 42 (1992) 171. | MR 1183841 | Zbl 0762.60059

[10] V. P. Belavkin and O. Melsheimer. A stochastic Hamiltonian approach for quantum jumps, spontaneous localizations, and continuous trajectories. Quantum Semiclass. Opt. 8 (1996) 167. | MR 1374516

[11] V. P. Belavkin and O. Melsheimer. Quantum diffusion, measurement and filtering. Probab. Theory Appl. 38 (1993) 742. | Zbl 0819.60046

[12] V. P. Belavkin and P. Staszewski. Quantum stochastic differential equation. J. Math. Phys. 41 (2000) 7220. | MR 1788571 | Zbl 0973.81063

[13] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR 1700749 | Zbl 0172.21201

[14] L. Bouten, M. Guţă and H. Maassen. Stochastic Schrödinger equations. J. Phys. A 37(9) (2004) 3189-3209. | MR 2042615 | Zbl 1074.81040

[15] P. Brémaud. Point Processes and Queues. Springer, New York, 1981. | MR 636252 | Zbl 0478.60004

[16] H. P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford Univ. Press, New York, 2002. | MR 2012610 | Zbl 1053.81001

[17] H. P. Breuer and F. Petruccione. Dissipative quantum systems in strong laser: Stochastic wave-function method and Floquet theory. Phys. Rev. A 55 (1997) 3101-3116.

[18] P. Cheridito, D. Filipović and M. Yor. Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Appl. Probab. 15(3) (2005) 1713-1732. | MR 2152242 | Zbl 1082.60034

[19] E. B. Davies. Quantum Theory of Open Systems. Academic Press, London, 1976. | MR 489429 | Zbl 0388.46044

[20] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. | MR 838085 | Zbl 1089.60005

[21] T. Fujiwara and H. Kunita. Limit theorems for stochastic difference-differential equations. Nagoya Math. J. 127 (1992) 83-116. | MR 1183654 | Zbl 0760.60034

[22] S. Haroche and J.-M. Raimond. Exploring the Quantum. Oxford Univ. Press, Oxford, 2006. | MR 2271425 | Zbl 1118.81001

[23] J. Jacod. Calcul stochastique et problèmes de martingales. Lecture Notes in Math. 714. Springer, Berlin, 1979. | MR 542115 | Zbl 0414.60053

[24] J. Jacod and P. Protter. Quelques remarques sur un nouveau type d'équations différentielles stochastiques. In Seminar on Probability, XVI, Lecture Notes in Math. 920 447-458. Springer, Berlin, 1982. | Numdam | MR 658706 | Zbl 0482.60056

[25] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin, 2003. | MR 1943877 | Zbl 1018.60002

[26] R. V. Kadison and J. R. Ringrose. Fundamentals of the Theory of Operator Algebras. Vol. I. Graduate Studies in Math. 15. Amer. Math. Soc., Providence, RI, 1997. | MR 1468229 | Zbl 0888.46039

[27] C. M. Mora and R. Rebolledo. Basic properties of non-linear stochastic Schrödinger equations driven by Brownian motions. Ann. Appl. Probab. 18 (2008) 591-619. | MR 2399706 | Zbl 1145.60036

[28] T. G. Kurtz and P. Protter. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19(3) (1991) 1035-1070. | MR 1112406 | Zbl 0742.60053

[29] C. Pellegrini. Existence, uniqueness and approximation for stochastic Schrödinger equation: The diffusive case. Ann. Probab. 36 (2008) 2332-2353. | MR 2478685 | Zbl 1167.60006

[30] C. Pellegrini. Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case. Preprint, 2008. Available at arXiv:0709.3713. | MR 2478685 | Zbl 1167.60006

[31] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Applications of Mathematics (New York) 21. Springer, Berlin, 2004. | MR 2020294 | Zbl 1041.60005

[32] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer, Berlin, 2006. | MR 2190038 | Zbl 1103.60005