Around Tsirelson’s equation, or: the evolution process may not explain everything. (English) Zbl 1328.60170

Summary: We present a synthesis of a number of developments which have been made around the celebrated Tsirelson’s equation from [B. S. Tsirel’son, Theory Probab. Appl. 20, 416–418 (1975); translation from Teor. Veroyatn. Primen. 20, 427–430 (1975; Zbl 0353.60061)], 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 \(G=\mathbb{T}\).


60J05 Discrete-time Markov processes on general state spaces
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60J50 Boundary theory for Markov processes
37H10 Generation, random and stochastic difference and differential equations


Zbl 0353.60061
Full Text: DOI arXiv Euclid


[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. · Zbl 1136.60036 · doi:10.1007/s00440-007-0076-z
[2] Birkhoff, G., A note on topological groups. Compositio Math. , 3:427-430, 1936. · Zbl 0015.00702
[3] Cirel’son, B. S., An example of a stochastic differential equation that has no strong solution. Teor. Verojatnost. i Primenen. , 20(2):427-430, 1975.
[4] Collins, H. S., Convergence of convolution iterates of measures. Duke Math. J. , 29:259-264, 1962. · Zbl 0129.09001 · doi:10.1215/S0012-7094-62-02926-5
[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. · Zbl 0144.39504 · doi:10.1007/BF00535358
[6] Dubins, L., Feldman, J., Smorodinsky, M., and Tsirelson, B., Decreasing sequences of \(\sigma\)-fields and a measure change for Brownian motion. Ann. Probab. , 24(2):882-904, 1996. · Zbl 0870.60078 · doi:10.1214/aop/1039639367
[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. · Zbl 0957.60064 · doi:10.1007/BFb0096518
[8] Engel, E. M. R. A., A road to randomness in physical systems , volume 71 of Lecture Notes in Statistics . Springer-Verlag, Berlin, 1992. · Zbl 0748.60102
[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. · Zbl 0942.22006
[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. · Zbl 0289.22011
[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. · Zbl 0121.35103 · doi:10.1137/1107031
[12] Hanson, D. L., On the representation problem for stationary stochastic processes with trivial tail field. J. Math. Mech. , 12:293-301, 1963. · Zbl 0139.34405
[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. · Zbl 1201.60072 · doi:10.1016/j.spa.2010.04.003
[14] Itô, K. and Nisio, M., On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ. , 4:1-75, 1964. · Zbl 0131.16402
[15] Kakutani, S., Über die Metrisation der topologischen Gruppen. Proc. Imp. Acad. , 12(4):82-84, 1936. · Zbl 0015.00701 · doi:10.3792/pia/1195580206
[16] Laurent, S., Further comments on the representation problem for stationary processes. Statist. Probab. Lett. , · Zbl 1187.60024 · doi:10.1016/j.spl.2009.12.015
[17] Rosenblatt, M., Stationary Markov chains and independent random variables. J. Math. Mech. , 9:945-949, 1960. · Zbl 0096.34004
[18] Stromberg, K., Probabilities on a compact group. Trans. Amer. Math. Soc. , 94:295-309, 1960. · Zbl 0109.10603 · doi:10.2307/1993313
[19] Tsirelson, B., Triple points: from non-Brownian filtrations to harmonic measures. Geom. Funct. Anal. , 7(6):1096-1142, 1997. · Zbl 0902.31004 · doi:10.1007/s000390050038
[20] Tsirelson, B., My drift, Citing works. .
[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.
[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.
[23] Yor, M., Tsirel’son’s equation in discrete time. Probab. Theory Related Fields , 91(2):135-152, 1992. · Zbl 0744.60033 · doi:10.1007/BF01291422
[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. · Zbl 0306.60049 · doi:10.1070/SM1974v022n01ABEH001689
[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. · Zbl 0481.60062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.