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 Around Tsirelson's equation, or: The evolution process may not explain everything
 Kouji Yano, Kyoto UniversityMarc Yor, Université Paris VI

 Abstract We present a synthesis of a number of developments which have been made around the celebrated Tsirelson's equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $$G$$, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $$G=\mathbb{T}$$.AMS 2000 subject classifications: Primary 60J05; secondary 60B15, 60J50, 37H10.Keywords: Tsirelson's equation, evolution process, extremal points, strong solution, uniqueness in law.
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Yano, Kouji, Yor, Marc, Around Tsirelson's equation, or: The evolution process may not explain everything, Probability Surveys, 12, (2015), 1-12 (electronic). DOI: 10.1214/15-PS256.

### References

[1]    Akahori, J., Uenishi, C., and Yano, K., Stochastic equations on compact groups in discrete negative time. Probab. Theory Related Fields, 140(3–4):569–593, 2008. MR2365485

[2]    Birkhoff, G., A note on topological groups. Compositio Math., 3:427–430, 1936. MR1556955

[3]    Cirelson, B. S., An example of a stochastic differential equation that has no strong solution. Teor. Verojatnost. i Primenen., 20(2):427–430, 1975. MR0375461

[4]    Collins, H. S., Convergence of convolution iterates of measures. Duke Math. J., 29:259–264, 1962. MR0137789

[5]    Csiszár, I., On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 5:279–295, 1966. MR0205306

[6]    Dubins, L., Feldman, J., Smorodinsky, M., and Tsirelson, B., Decreasing sequences of σ-fields and a measure change for Brownian motion. Ann. Probab., 24(2):882–904, 1996. MR1404533

[7]    Émery, M. and Schachermayer, W., A remark on Tsirelson’s stochastic differential equation. In Séminaire de Probabilités, XXXIII, volume 1709 of Lecture Notes in Math., pages 291–303. Springer, Berlin, 1999. MR1768003

[8]    Engel, E. M. R. A., A road to randomness in physical systems, volume 71 of Lecture Notes in Statistics. Springer-Verlag, Berlin, 1992. MR1177596

[9]    Furstenberg, H., Stiffness of group actions. Lie groups and ergodic theory (Mumbai, 1996), 105–117, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998. MR1699360

[10]    Furstenberg, H., Boundary theory and stochastic processes on homogeneous spaces. Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), 193–229. Amer. Math. Soc., Providence, R.I., 1973. MR0352328

[11]    Girsanov, I. V., An example of non-uniqueness of the solution of the stochastic equation of K. Itô. Theory Probab. Appl., 7:325–331, 1962.

[12]    Hanson, D. L., On the representation problem for stationary stochastic processes with trivial tail field. J. Math. Mech., 12:293–301, 1963. MR0146896

[13]    Hirayama, T. and Yano, K., Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups. Stochastic Process. Appl., 120(8):1404–1423, 2010. MR2653259

[14]    Itô, K. and Nisio, M., On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ., 4:1–75, 1964. MR0177456

[15]    Kakutani, S., Über die Metrisation der topologischen Gruppen. Proc. Imp. Acad., 12(4):82–84, 1936. MR1568424

[16]    Laurent, S., Further comments on the representation problem for stationary processes. Statist. Probab. Lett., to appear. MR2595135

[17]    Rosenblatt, M., Stationary Markov chains and independent random variables. J. Math. Mech., 9:945–949, 1960. MR0166839

[18]    Stromberg, K., Probabilities on a compact group. Trans. Amer. Math. Soc., 94:295–309, 1960. MR0114874

[19]    Tsirelson, B., Triple points: from non-Brownian filtrations to harmonic measures. Geom. Funct. Anal., 7(6):1096–1142, 1997. MR1487755

[20]    Tsirelson, B., My drift, Citing works. http://www.math.tau.ac.il/~tsirel/Research/mydrift/citing.html.

[21]    Vershik, A. M., Decreasing sequences of measurable partitions and their applications. Dokl. Akad. Nauk SSSR, 193(4):748–751, 1970; English transl. in Soviet Math. Dokl. 11(4):1007–1011, 1970. MR1819177

[22]    Wiener, N., Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications. The Technology Press of the Massachusetts Institute of Technology, Cambridge, Mass, 1949. MR0031213

[23]    Yor, M., Tsirel’son’s equation in discrete time. Probab. Theory Related Fields, 91(2):135–152, 1992. MR1147613

[24]    Zvonkin, A. K., A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. (N.S.), 93(135):129–149, 152, 1974. MR0336813

[25]    Zvonkin, A. K. and Krylov, N. V., Strong solutions of stochastic differential equations. In Proceedings of the School and Seminar on the Theory of Random Processes (Druskininkai, 1974), Part II (Russian), pages 9–88. Inst. Fiz. i Mat. Akad. Nauk Litovsk. SSR, Vilnius, 1975. MR0426154

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Probability Surveys. ISSN: 1549-5787