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 Probability Surveys > Vol. 11 (2014) open journal systems 

Reciprocal processes. A measure-theoretical point of view

Christian Léonard, Paris Ouest University
Sylvie Rœlly, Potsdam University
Jean-Claude Zambrini, GFM, Lisbon University

The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal process.
The structures of Markov and reciprocal processes are recalled with emphasis on their time-symmetries. A review of the main properties of the reciprocal processes is presented. Our measure-theoretical approach allows for a unified treatment of the diffusion and jump processes.

Keywords: Markov process, reciprocal process, Markov bridge, time-symmetry, entropy minimization.

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Léonard, Christian, Rœlly, Sylvie, Zambrini, Jean-Claude, Reciprocal processes. A measure-theoretical point of view, Probability Surveys, 11, (2014), 237-269 (electronic). DOI: 10.1214/13-PS220.


[1]    Aebi, R. (1996). Schrödinger Diffusion Processes. Birkhäuser. MR1391719

[2]    Arnold, V. (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 1, 319–361. MR0202082

[3]    Bernstein, S. (1932). Sur les liaisons entre les grandeurs aléatoires. Verhand. Internat. Math. Kongr. Zürich, Band I.

[4]    Beurling, A. (1960). An automorphism of product measures. Ann. of Math. 72, 189–200. MR0125424

[5]    Brenier, Y. (1989). The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Amer. Math. Soc. 2, 2, 225–255. MR0969419

[6]    Chaumont, L. and Uribe-Bravo, G. (2011). Markovian bridges: Weak continuity and pathwise constructions. Ann. Probab. 39, 609–647. MR2789508

[7]    Chay, S. (1972). On quasi-Markov random fields. J. Multivariate Anal. 2, 14–76. MR0303589

[8]    Chung, K. (1968). A Course in Probability Theory. Harcourt, Brace & World, Inc., New York. MR0229268

[9]    Chung, K. and Walsh, J. (2005). Markov Processes, Brownian Motion and Time Symmetry. Lecture Notes in Mathematics. Springer Verlag, New York. MR2152573

[10]    Chung, K. and Zambrini, J.-C. (2008). Introduction to Random Time and Quantum Randomness. World Scientific. MR1999000

[11]    Clark, J. (1991). A local characterization of reciprocal diffusions. Applied Stoch. Analysis 5, 45–59. MR1108416

[12]    Conforti, G., Dai Pra, P., and Rœlly, S., Reciprocal classes of jump processes. Preprint. http://opus.kobv.de/ubp/volltexte/2014/7077/pdf/premath06.pdf.

[13]    Conforti, G. and Léonard, C., Reciprocal classes of random walks on graphs. In preparation.

[14]    Conforti, G., Léonard, C., Murr, R., and Rœlly, S., Bridges of Markov counting processes. Reciprocal classes and duality formulas. Preprint, arXiv:1408.1332.

[15]    Courrège, P. and Renouard, P. (1975). Oscillateurs harmoniques mesures quasi-invariantes sur c(, ) et théorie quantique des champs en dimension 1. In Astérisque. Vol. 22–23. Société Mathématique de France. MR0496165

[16]    Cruzeiro, A., Wu, L., and Zambrini, J.-C. (2000). Bernstein processes associated with a Markov process. In Stochastic Analysis and Mathematical Physics, ANESTOC’98. Proceedings of the Third International Workshop, R. Rebolledo, Ed., Trends in Mathematics. Birkhäuser, Boston, 41–71. MR1764785

[17]    Cruzeiro, A. and Zambrini, J.-C. (1991). Malliavin calculus and Euclidean quantum mechanics, I. J. Funct. Anal. 96, 1, 62–95. MR1093507

[18]    Dang Ngoc, N. and Yor, M. (1978). Champs markoviens et mesures de Gibbs sur . Ann. Sci. Éc. Norm. Supér., 29–69. MR0504421

[19]    Doob, J. (1953). Stochastic Processes. Wiley. MR0058896

[20]    Doob, J. (1957). Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. Fr. 85, 431–458. MR0109961

[21]    Dynkin, E. (1961). Theory of Markov Processes. Prentice-Hall Inc.

[22]    Fitzsimmons, P., Pitman, J., and Yor, M. (1992). Markovian bridges: Construction, Palm interpretation, and splicing. Progr. Probab. 33, 101–134. MR1278079

[23]    Föllmer, H. (1988). Random fields and diffusion processes, in École d’été de Probabilités de Saint-Flour XV-XVII-1985-87. Lecture Notes in Mathematics, Vol. 1362. Springer, Berlin. MR0983373

[24]    Föllmer, H. and Gantert, N. (1997). Entropy minimization and Schrödinger processes in infinite dimensions. Ann. Probab. 25, 2, 901–926. MR1434130

[25]    Georgii, H.-O. (2011). Gibbs measures and phase transitions. In Studies in Mathematics, Second ed. Vol. 9. Walter de Gruyter. MR2807681

[26]    Jamison, B. (1970). Reciprocal processes: The stationary Gaussian case. Ann. Math. Statist. 41, 1624–1630. MR0267637

[27]    Jamison, B. (1974). Reciprocal processes. Z. Wahrsch. verw. Geb. 30, 65–86. MR0359016

[28]    Jamison, B. (1975). The Markov processes of Schrödinger. Z. Wahrsch. verw. Geb. 32, 4, 323–331. MR0383555

[29]    Kallenberg, O. (1981). Splitting at backward times in regenerative sets. Annals Probab. 9, 781–799. MR0628873

[30]    Kolmogorov, A. (1936). Zur Theorie der Markoffschen Ketten. Mathematische Annalen 112.

[31]    Krener, A. (1988). Reciprocal diffusions and stochastic differential equations of second order. Stochastics 24, 393–422. MR0972972

[32]    Kullback, S. and Leibler, R. (1951). On information and sufficiency. Annals of Mathematical Statistics 22, 79–86. MR0039968

[33]    Léonard, C., Some properties of path measures. To appear in Séminaire de probabilités de Strasbourg, vol. 46. Preprint, arXiv:1308.0217.

[34]    Léonard, C. (2014). A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34, 4, 1533–1574. MR3121631

[35]    Meyer, P.-A. (1967). Processus de Markov. Lecture Notes in Mathematics, Vol. 26. Springer Verlag, New York. MR0219136

[36]    Murr, R. (2012). Reciprocal classes of Markov processes. An approach with duality formulae. Ph.D. thesis, Universität Potsdam, http://opus.kobv.de/ubp/volltexte/2012/6301.

[37]    Nagasawa, M. (1993). Schrödinger Equations and Diffusions Theory. Monographs in Mathematics, Vol. 86. Birkhäuser. MR1227100

[38]    Nelson, E. (1967). Dynamical Theories of Brownian Motion. Princeton University Press. Second edition (2001) at: http://www.math.princeton.edu/nelson/books.html. MR0214150

[39]    Osada, H. and Spohn, H. (1999). Gibbs measures relative to Brownian motion. Annals Probab. 27, 1183–1207. MR1733145

[40]    Privault, N. and Zambrini, J.-C. (2004). Markovian bridges and reversible diffusions with jumps. Ann. Inst. H. Poincaré. Probab. Statist. 40, 599–633. MR2086016

[41]    Rœlly, S. (2014). Reciprocal processes. A stochastic analysis approach. In Modern Stochastics with Applications. Optimization and Its Applications, Vol. 90. Springer, 53–67.

[42]    Rœlly, S. and Thieullen, M. (2004). A characterization of reciprocal processes via an integration by parts formula on the path space. Probab. Theory Related Fields 123, 97–120. MR1906440

[43]    Rœlly, S. and Thieullen, M. (2005). Duality formula for the bridges of a brownian diffusion: Application to gradient drifts. Stochastic Processes and Their Applications 115, 1677–1700. MR2165339

[44]    Royer, G. and Yor, M. (1976). Représentation intégrale de certaines mesures quasi-invariantes sur c(); mesures extrémales et propriété de Markov. Ann. Inst. Fourier 26, 7–24. MR0447517

[45]    Schrödinger, E. (1931). Über die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144, 144–153.

[46]    Schrödinger, E. (1932). Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. H. Poincaré 2, 269–310. MR1508000

[47]    Thieullen, M. (1993). Second order stochastic differential equations and non-gaussian reciprocal diffusions. Probab. Theory Related Fields 97, 231–257. MR1240725

[48]    Thieullen, M. (2002). Reciprocal diffusions and symmetries of parabolic PDE: The nonflat case. Potential Analysis 16, 1, 1–28. MR1880345

[49]    Thieullen, M. and Zambrini, J.-C. (1997a). Probability and quantum symmetries. I. The theorem of Noether in Schrödinger’s Euclidean quantum mechanics. Ann. Inst. H. Poincaré Phys. Théor. 67, 297–338. MR1472821

[50]    Thieullen, M. and Zambrini, J.-C. (1997b). Symmetries in the stochastic calculus of variations. Probab. Theory Related Fields 107, 3, 401–427. MR1440139

[51]    van Putten, C. and van Schuppen, J. (1985). Invariance properties of the conditional independence relation. Ann. Probab. 13, 934–945. MR0799429

[52]    Vuillermot, P. and Zambrini, J.-C. (2012). Bernstein diffusions for a class of linear parabolic PDEs. Journal of Theor. Probab., 1–44.

[53]    Wentzell, A. (1981). A course in the theory of stochastic processes. Mc Graw-Hill.

[54]    Zambrini, J.-C., The research program of stochastic deformation (with a view toward geometric mechanics). To be published in Stochastic Analysis. A Series of Lectures. Birkhaüser. Preprint, arXiv:1212.4186.

[55]    Zambrini, J.-C. (1986). Variational processes and stochastic version of mechanics. Journal of Mathematical Physics 27, 2307–2330. MR0854761

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