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 Probability Surveys > Vol. 1 (2004) open journal systems 

Nonclassical stochastic flows and continuous products

Boris Tsirelson, Tel Aviv University

Contrary to the classical wisdom, processes with independent values (defined properly) are much more diverse than white noises combined with Poisson point processes, and product systems are much more diverse than Fock spaces. This text is a survey of recent progress in constructing and investigating nonclassical stochastic flows and continuous products of probability spaces and Hilbert spaces.

AMS 2000 subject classifications: Primary 60G20; secondary 46L53.

Keywords: stochastic flows, continuous products, noise, stability, sensitivity.

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Tsirelson, Boris, Nonclassical stochastic flows and continuous products, Probability Surveys, 1, (2004), 173-298 (electronic).


[1]   L. Accardi, On square roots of measures, In: Proc. Internat. School of Physics “Enrico Fermi”, Course LX, North-Holland, pp. 167–189 (1976). MR0626536

[2]   H. Araki and E.J. Woods, Complete Boolean algebras of type I factors, Publications of the Research Institute for Mathematical Sciences, Kyoto Univ., Series A, 2(2), 157–242 (1966). MR0203497

[3]   R.A. Arratia, Coalescing Brownian motions on the line, Ph. D. Thesis, Univ. of Wisconsin, Madison, 1979.

[4]   W. Arveson, Noncommutative dynamics and E-semigroups, Springer, New York, 2003. MR1978577

[5]   M.T. Barlow, M. Émery, F.B. Knight, S. Song and M. Yor, Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes, Lect. Notes in Math 1686 (Séminaire de Probabilités XXXII), Springer, Berlin, 264–305 (1998). MR1655299

[6]   P. Baxendale, Brownian motions in the diffeomorphism group I, Compositio Mathematica 53:1, 19–50 (1984). MR0762306

[7]   H. Becker and A. S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series 232, Cambridge University Press, 1996. MR1425877

[8]   I. Benjamini, G. Kalai and O. Schramm, Noise sensitivity of Boolean functions and applications to percolation, arXiv:math.PR/9811157v2. Inst. Hautes Études Sci. Publ. Math. no. 90, 5–43 (1999). MR1813223

[9]   B. Bhat, Cocycles of CCR flows, Memoirs Amer. Math. Soc. 149 (709), 114 pp (2001). MR1804156.

[10]   B.V.R. Bhat and R. Srinivasan, On product systems arising from sum systems, arXiv:math.OA/0405276v1.

[11]   E.B. Davies, Quantum theory of open systems, Academic Press, London, 1976. MR0489429

[12]   M. Émery and W. Schachermayer, A remark on Tsirelson’s stochastic differential equation, Lecture Notes in Math. 1709 (Séminaire de Probabilités XXXIII), Springer, Berlin, 291–303 (1999). MR1768002

[13]   J. Feldman, Decomposable processes and continuous products of probability spaces, J. Funct. Anal. 8, 1–51 (1971). MR0290436

[14]   L.R.G. Fontes, M. Isopi, C.M. Newman, K. Ravishankar, The Brownian web: characterization and convergence, arXiv:math.PR/0311254v1. Annals of Probability 32:4 (2004).

[15]   E. Glasner, B. Tsirelson and B. Weiss, The automorphism group of the Gaussian measure cannot act pointwise, arXiv:math.DS/0311450v2. Israel J. Math. (to appear).

[16]   T.E. Harris, Coalescing and noncoalescing stochastic flows in ℝ1, Stochastic Processes and their Applications 17:2, 187–210 (1984). MR0751202

[17]   A.S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Math. 156, Springer, New York, 1995. MR1321597

[18]   A.S. Kechris, New directions in descriptive set theory, The Bulletin of Symbolic Logic 5:2, 161–174 (1999). MR1791302

[19]   Y. Le Jan, S. Lemaire, Products of Beta matrices and sticky flows, arXiv:math.PR/0307106v3. Probability Theory and Related Fields 130:1, 109–134 (2004).

[20]   Y. Le Jan, O. Raimond, Flows, coalescence and noise, arXiv:math.PR/0203221v4. The Annals of Probability 32:2, 1247–1315 (2004). MR2060298

[21]   Y. Le Jan, O. Raimond, Sticky flows on the circle and their noises, Probability Theory and Related Fields 129:1, 63–82 (2004). MR2052863

[22]   V. Liebscher, Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces, arXiv:math.PR/0306365v1.

[23]   G. Link, Representation theorems of the de Finetti type for (partially) symmetric probability measures, In: Studies in inductive logic and probability, vol. II, 207–231, Univ. of California Press, Berkeley, 1980. MR0587992

[24]   J. Neveu, Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, 1965. MR0198505

[25]   R.T. Powers, A nonspatial continuous semigroup of *-endomorphisms of 𝔅(), Publications of the Research Institute for Mathematical Sciences, Kyoto Univ., 23:6, 1053–1069 (1987). MR0935715

[26]   R.T. Powers, New examples of continuous spatial semigroups of *-endomorphisms of 𝔅(), Internat. J. Math. 10:2, 215–288 (1999). MR1687149

[27]   M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Second edition, Academic Press, New York, 1980. MR0751959

[28]   D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Second edition. Springer, Berlin, 1994. MR1303781

[29]   O. Schramm and B. Tsirelson, Trees, not cubes: hypercontractivity, cosiness, and noise stability, Electronic Communications in Probability 4, 39–49 (1999). MR1711603

[30]   A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., 50:12, 1261–1286 (1997). MR1476315

[31]   A.V. Skorokhod, Random linear operators, Mathematics and its Applications (Soviet Series), D. Reidel Publ., Dordrecht, 1984. MR0733994

[32]   S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation, arXiv:math.PR/0109120v2. Mathematical Research Letters 8:5/6, 729–744 (2001). MR1879816

[33]   F. Soucaliuc, B. Tóth and W. Werner, Reflection and coalescence between independent one-dimensional Brownian paths, Ann. Inst. H. Poincaré Probab. Statist. 36:4, 509–545 (2000). MR1785393

[34]   B. Tóth and W. Werner, The true self-repelling motion, Probab. Theory Related Fields 111:3, 375–452 (1998). MR1640799

[35]   B. Tsirelson, Unitary Brownian motions are linearizable, arXiv:math.PR/9806112v1.

[36]   B. Tsirelson, Noise sensitivity on continuous products: an answer to an old question of J. Feldman, arXiv:math.PR/9907011v1.

[37]   B. Tsirelson, From random sets to continuous tensor products: answers to three questions of W. Arveson, arXiv:math.FA/0001070v1.

[38]   B. Tsirelson, Spectral densities describing off-white noises, arXiv:math.FA/0104027v1. Ann. Inst. H. Poincare Probab. Statist., 38:6 (2002), 1059–1069. MR1955353

[39]    B. Tsirelson, Non-isomorphic product systems, arXiv:math.FA/0210457v2. In: Advances in Quantum Dynamics (eds. G. Price et al), Contemporary Mathematics 335, AMS, pp. 273–328 (2003). MR2029632 [Note: the numbers of theorems, equations etc. refer to the arXiv version. In the AMS version they differ (because of a different LATEX style) as follows (arXiv/AMS): 2.2/2.1; 2.4/2.3; 2.5/2.4; 2.6/2.5; 2.9/2.6; 5.3/5.3; 7.3/7.3; 8.5/8.1; 9.7/9.6; (9.11)/(29); (9.12)/(30); (9.13)/(31); 10.2/10.1; 10.3/10.2; 11.3/11.3; 13.10/13.5; 13.11/13.6.]

[40]   B. Tsirelson, Scaling limit, noise, stability, arXiv:math.PR/0301237v1. Lect. Notes in Math 1840 (St. Flour XXXII), Springer, Berlin, 1–106 (2004). MR2079671 [Note: the numbers of subsections, theorems etc. refer to the arXiv version. In the Springer version they differ (because of a different LATEX style) as follows (arXiv/Springer). Subsections: 1a/1.1; 5b/5.2; 6b/6.2; 7d/7.4; 8b/8.2; 8d/8.4. Theorems etc.: 1d1/1.9; (3d3)/(3.11); 3d6/3.20; 3d11/3.25; 3d12/3.26; 3e3/3.28; 5b4/5.5; 5b5/5.6; 5b11/5.10; 6a3/6.2; 6a4/6.3; 6b1/6.6; 6b2/6.7; . . . ; 6b12/6.17; 6c4/6.18; 6c7/6.21; 8a1/8.1; 8a2/8.2; 8d3/8.8.]

[41]   B. Tsirelson, On automorphisms of type II Arveson systems (probabilistic approach), arXiv:math.OA/0411062v1.

[42]   B.S. Tsirelson and A.M. Vershik, Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations, Reviews in Mathematical Physics 10:1, 81–145 (1998). MR1606855

[43]   J. Warren, On the joining of sticky Brownian motion, Lecture Notes in Math. 1709 (Séminaire de Probabilités XXXIII), Springer, Berlin, 257–266 (1999). MR1767999

[44]   J. Warren, Splitting: Tanaka’s SDE revisited, arXiv:math.PR/9911115v1.

[45]   J. Warren, The noise made by a Poisson snake, Electronic Journal of Probability 7:21, 1–21 (2002). MR1943894

[46]   J. Warren and S. Watanabe, On spectra of noises associated with Harris flows, arXiv:math.PR/0307287v1. Advanced Studies in Pure Mathematics 41, 351–373 (2004). MR2083719

[47]   S. Watanabe, Stochastic flow and noise associated with the Tanaka stochastic differential equation, Ukrainian Math. J. 52:9, 1346–1365 (2001) (transl). MR1816931

[48]   S. Watanabe, A simple example of black noise, Bull. Sci. Math. 125:6/7, 605–622 (2001). MR1869993

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

[50]   K. Yosida, On Brownian motion in a homogeneous Riemannian space, Pacific J. Math. 2, 263–270 (1952). MR0050817

[51]   J. Zacharias, Continuous tensor products and Arveson’s spectral C*-algebras, Memoirs Amer. Math. Soc. 143 (680), 118 pp (2000). MR1643205

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