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On the notion(s) of duality for Markov processes. (English) Zbl 1292.60077
Summary: We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.

MSC:
60J25 Continuous-time Markov processes on general state spaces
46N30 Applications of functional analysis in probability theory and statistics
47D07 Markov semigroups and applications to diffusion processes
60J05 Discrete-time Markov processes on general state spaces
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[1] Aizenman, M., Nachtergaele, B., Geometric aspects of quantum spin states. Comm. Math. Phys. 164(1):17-63 (1994). · Zbl 0799.58087 · doi:10.1007/BF02108805
[2] Alkemper, R., Hutzenthaler, M., Graphical representation of some duality relations in stochastic population models. Electr. Comm. Prob. 12:206-220 (2007). · Zbl 1129.60093 · eudml:128368
[3] Asmussen, S., Applied probability and queues . 2nd edition, Springer-Verlag, New York, 2003. · Zbl 1029.60001
[4] Asmussen, S., Sigman, K., Monotone stochastic recursion and their duals. Prog. Eng. Inf. Sci. 10:1-20 (1996). · Zbl 1095.60519 · doi:10.1017/S0269964800004137
[5] Athreya, S., Swart, J., Systems of branching, annihilating, and coalescing particles. Electron. Journ. Prob. 17(80):1-32 (2012). · Zbl 1252.82064
[6] Athreya, S., Tribe, R., Uniqueness for a class of one-dimensional stochastic PDEs using moment duality. The Annals of Probability 28(4):1711-1734 (2000). · Zbl 1044.60048 · doi:10.1214/aop/1019160504
[7] Bertoin, J., Le Gall, J.-F., Stochastic flows associated to coalescent processes. Prob. Theory Rel. Fields 126(2):261-288 (2003). · Zbl 1023.92018 · doi:10.1007/s00440-003-0264-4
[8] Birkner, M., Möhle, J. M., Steinrücken, M., Tams, J., A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks. Alea 6:25-61 (2009). · Zbl 1162.60342 · alea.impa.br · arxiv:0808.0412
[9] Blumenthal, R. M., Getoor, R. K., Markov processes and potential theory . Pure and Applied Mathematics, vol. 29, Academic Press, New York London, 1968. · Zbl 0169.49204 · www.sciencedirect.com
[10] Bogachev, V. I., Measure theory, Volume 2 . Springer-Verlag, Berlin Heidelberg, 2007. · Zbl 1120.28001
[11] Bratteli, O., Robinson, D. W., Operator algebras and quantum statistical mechanics , vol. 1. 2nd edition, Springer, 1987. · Zbl 0905.46046
[12] Carmona, Ph., Petit, F., Yor, M., Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoamericana 14(2):311-367 (1998). · Zbl 0919.60074 · doi:10.4171/RMI/241 · eudml:39554
[13] Chung, K. L., Walsh, J. B., Markov processes, Brownian motion, and time symmetry . 2nd edition, Springer, New York, 2005. · Zbl 1082.60001
[14] Cox, J. Th., Rösler, U., A duality relation for entrance and exit laws for Markov processes. Stochastic Process. Appl. 16(2):141-156 (1984). · Zbl 0523.60068 · doi:10.1016/0304-4149(84)90015-2
[15] Clifford, P., Sudbury, A., A sample path proof of the duality for stochastically monotone Markov processes. Ann. Probab. 13(2):558-565 (1985). · Zbl 0563.60062 · doi:10.1214/aop/1176993008
[16] Dawson, D., Greven, A., Spatial Fleming-Viot models with selection and mutation . Lecture Notes in Mathematics, vol. 2092, Springer, 2014. · Zbl 1295.92008
[17] Dawson, D., Kurtz, T., Applications of duality to measure-valued diffusion processes. In Advances in filtering and optimal stochastic control (Cocoyoc, 1982) . Lecture Notes in Control and Inform. Sci., vol. 42, pages 91-105, Springer, 1982. · Zbl 0496.60057 · doi:10.1007/BFb0004528
[18] De Masi, A., Presutti, E., Mathematical methods for hydrodynamic limits . Lecture Notes in Mathematics, vol. 1501, Springer, Berlin, 1991. · Zbl 0754.60122 · doi:10.1007/BFb0086457
[19] Dette, H., Fill, J. A., Pitman, J., Studden, W. J., Wall and Siegmund duality relations for birth and death chains with reflecting barrier. J. Theor. Probab. 10:349-374 (1997). · Zbl 0894.60076 · doi:10.1023/A:1022660400024
[20] Diaconis, P., Fill, J. A., Strong stationary times via a new form of duality. Ann. Probab. 18:1483-1522 (1990). · Zbl 0723.60083 · doi:10.1214/aop/1176990628
[21] Doi, M., Second quantization representation for classical many-particle system. J. Phys. A: Math. Gen. 9:1465-1477 (1976).
[22] Donnelly, P., Kurtz, Th. G., A countable representation of the Fleming-Viot measure valued diffusion. Ann. Probab. 24(2):698-742 (1996). · Zbl 0869.60074 · doi:10.1214/aop/1039639359
[23] Donnelly, P., Kurtz, Th. G., Particle representations for measure-valued population models. Ann. Probab. 27(1):166-205 (1999). · Zbl 0956.60081 · doi:10.1214/aop/1022677258
[24] Diaconis, P., Miclo, L., On times to quasi-stationarity for birth and death processes. J. Theoret. Probab. 22(3):558-586 (2009). · Zbl 1186.60086 · doi:10.1007/s10959-009-0234-6
[25] Dynkin, E. B., Markov Processes . Springer, 1965.
[26] Etheridge, A., Evolution in fluctuating populations. In Mathematical statistical physics , pages 489-545, Elsevier B. V., Amsterdam, 2006. · Zbl 1417.92111
[27] Etheridge, A., Some mathematical models from population genetics . Lecture Notes in Mathematics, Springer, 2011. · Zbl 1320.92003
[28] Ethier, S. N., Krone, S. M., Comparing Fleming-Viot and Dawson-Watanabe processes. Stoch. Proc. Appl. 60:401-421 (1995). · Zbl 0845.60042 · doi:10.1016/0304-4149(95)00056-9
[29] Ethier, S. N., Kurtz, Th. G., Markov processes, characterization and convergence . Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1986. · Zbl 0592.60049
[30] Feller, W., An introduction to probability theory and its applications , vol. 2. 2nd edition, Wiley, 1971. · Zbl 0219.60003
[31] Fill, J. A., Strong stationary duality for continuous-time Markov chains. I. Theory. J. Theoret. Probab. 5(1):45-70 (1992). · Zbl 0746.60075 · doi:10.1007/BF01046778
[32] Getoor, R. K., Duality theory for Markov processes: Part I. 2010). arXiv: · arxiv.org
[33] Giardina, C., Kurchan, J., Redig, F., Vafayi, K., Duality and hidden symmetries in interacting particle systems. J. Stat. Phys. 135:25-55 (2009). · Zbl 1173.82020 · doi:10.1007/s10955-009-9716-2
[34] Goldschmidt, Ch., Ueltschi, D., Windridge, P., Quantum Heisenberg models and their probabilistic representations. Entropy and the quantum II. Contemp. Math. 552:177-224 (2011). · Zbl 1244.82010 · doi:10.1090/conm/552/10917
[35] Griffeath, D., Additive and cancellative interacting particle systems . Lecture Notes in Mathematics, vol. 724, Springer, 1979. · Zbl 0412.60095 · doi:10.1007/BFb0067306
[36] Gwa, L.-H., Spohn, H., Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68(6):725-728 (1992). · Zbl 0969.82526 · doi:10.1103/PhysRevLett.68.725
[37] Harris, T. E., Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 (3):355-3778 (1978). · Zbl 0378.60106 · doi:10.1214/aop/1176995523
[38] Hewitt, E., Ross, K. A., Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups . Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer, New York Berlin, 1970. · Zbl 0213.40103
[39] Hoffman, K., Banach spaces of analytic functions . Prentice-Hall Series in Modern Analysis, 1962. · Zbl 0117.34001
[40] Holley, R. A., Liggett, T. M., Ergodic theorems for weakly interacting infinite systems and the Voter model. Ann. Probab. 3(4):643-663 (1975). · Zbl 0367.60115 · doi:10.1214/aop/1176996306
[41] Holley, R., Stroock, D., Williams, D., Applications of dual processes to diffusion theory. Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976) , 23-36 (1977). · Zbl 0382.60081
[42] Holley, R., Stroock, D., Dual processes and their application to infinite interacting systems. Adv. in Math. 32:149-174 (1979). · Zbl 0459.60097 · doi:10.1016/0001-8708(79)90040-9
[43] Hunt, G. A., Markov processes and potentials III. Illinois J. Math. 2:151-213 (1958). · Zbl 0100.13804
[44] Huillet, T., Martinez, S., Duality and intertwining for discrete Markov kernels: relations and examples. Adv. in Appl. Probab. 43(2):437-460 (2011). · Zbl 1225.60127 · doi:10.1239/aap/1308662487 · projecteuclid.org
[45] Jansen, S., Kurt, N., Pathwise construction of certain moment dualities and application to population models with balancing selection. Electron. Comm. Probab. 18 , paper no. 14 (2013). · Zbl 1335.60183 · doi:10.1214/ECP.v18-2194
[46] Karlin, S., McGregor, J., The classification of birth and death processes. Trans. Amer. Math. Soc. 86:366-400 (1957). · Zbl 0091.13802 · doi:10.2307/1993021
[47] Kirillov, A., Jr., An introduction to Lie groups and Lie algebras . Cambridge Studies in Advanced Mathematics, vol. 113, Cambridge University Press, Cambridge, 2008. · Zbl 1153.17001
[48] Klebaner, F., Rösler, U., Sagitov, S., Transformations of Galton-Watson proceses and linear fractional reproduction. Adv. Appl. Probab. 39:1036-1053 (2007). · Zbl 1139.60038 · doi:10.1239/aap/1198177238
[49] Lévy, P., Processus stochastiques et mouvement Brownien . Gauthier-Villars, Paris, 1948. · Zbl 0034.22603
[50] Liggett, Th. M., Interacting particle systems . Classics in Mathematics, Springer-Verlag, Berlin, 2005. Reprint of the 1985 original. · Zbl 1103.82016
[51] Lindley, D., The theory of queues with a single server. Proc. Cambridge Philos. Soc. 48:277-289 (1952). · Zbl 0046.35501 · doi:10.1017/S0305004100027638
[52] Lloyd, P., Sudbury, A., Donnelly, P., Quantum operators in classical probability theory. I. “Quantum spin” techniques and the exclusion model of diffusion. Stochastic Process. Appl. 61(2):205-221 (1996). · Zbl 0847.60093 · doi:10.1016/0304-4149(96)84552-2
[53] Möhle, M., The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5(5):761-777 (1999). · Zbl 0942.92020 · doi:10.2307/3318443
[54] Möhle, M., Duality and cones of Markov processes and their semigroups, preprint (2011).
[55] Moran, P. A. P., Random processes in genetics. Proc. Cambridge Philos. Soc. 54:60-71 (1958). · Zbl 0091.15701 · doi:10.1017/S0305004100033193
[56] Mytnik, L., Weak uniqueness for the heat equation with noise. Ann. Probab. 26,(3):968-984 (1998). · Zbl 0935.60045 · doi:10.1214/aop/1022855740
[57] Peliti, L., Path integral approach to birth-death processes on a lattice. J. Phys. France 46:1469-1483 (1985).
[58] Phelps, R. R., Lectures on Choquet’s theorem . 2nd edition, Springer, 2001. · Zbl 0997.46005
[59] Reed, M., Simon, B., Modern methods of mathematical physics, Vol. 1: Functional analysis . Revised and enlarged edition, Academic Press Inc., 1980. · Zbl 0459.46001
[60] Rogers, L. C. G., Williams, D., Diffusions, Markov processes and martingales , Vol. I. 2nd edition, Cambridge University Press, 1994. · Zbl 0826.60002
[61] Saloff-Coste, L., Probability on groups: Random walks and invariant diffusions. Notices Amer. Math. Soc. 48(9):968-977 (2001). · Zbl 0987.60018
[62] Schütz, G., Sandow, S., Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems. Phys. Rev. E 49:2726-2741 (1994).
[63] Siegmund, D., The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Probab. 4:914-924 (1976). · Zbl 0364.60109 · doi:10.1214/aop/1176995936
[64] Sudbury, A., Dual families of interacting particle systems on graphs. J. Theor. Probab. 13(3):695-716 (2000). · Zbl 0968.60096 · doi:10.1023/A:1007806427774
[65] Sudbury, A., Lloyd, P., Quantum operators in classical probability theory. II. The concept of duality in interacting particle systems. Ann. Probab. 23(4):1816-1830 (1995). · Zbl 0853.60079 · doi:10.1214/aop/1176987804
[66] Sudbury, A., Lloyd, P., Quantum operators in classical probability theory. IV. Quasi-duality and thinnings of interacting particle systems. Ann. Probab. 25(1):96-114 (1997). · Zbl 0873.60075 · doi:10.1214/aop/1024404280
[67] Spitzer, F., Interaction of Markov processes. Advances in Math. 5:246-290 (1970). · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4
[68] Swart, J., Duals and thinnings of some relatives of the contact process. 2006).
[69] Swart, J., Intertwining of birth-and-death processes. Kybernetika 47(1):1-14 (2011). · Zbl 1221.60125 · www.kybernetika.cz · eudml:196582
[70] Swart, J., Duality and intertwining of Markov chains. Lecture notes for the ALEA in Europe School, Marseille 2013. Available at . · staff.utia.cas.cz
[71] Tóth, B., Improved lower bound on the thermodynamic pressure of the spin 1=2 Heisenberg ferromagnet. Lett. Math. Phys. 28(1):75-84 (1993). · Zbl 0772.60103 · doi:10.1007/BF00739568
[72] Vervaat, W., Algebraic duality of Markov processes. In Mark Kac Seminar on Probability and Physics. Syllabus 1985-1987 (Amsterdam, 1985-1987) . CWI Syllabi, vol. 17, pages 61-69, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1988. · Zbl 0668.60063
[73] Werner, D., Funktionalanalysis . 3rd edition, Springer, 2000.
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