Stochastic Bäcklund Transformations

  • Neil O’Connell
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


How does one introduce randomness into a classical dynamical system in order to produce something which is related to the ‘corresponding’ quantum system? We consider this question from a probabilistic point of view, in the context of some integrable Hamiltonian systems.


Stochastic Differential Equation Infinitesimal Generator Toda Lattice Martingale Problem Toda Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Thanks to Simon Ruijsenaars for valuable discussions and comments on an earlier draft, and Mark Adler and Tom Kurtz for helpful correspondence. Thanks also to the anonymous referee for helpful comments and suggestions.


  1. 1.
    F. Baudoin, Further exponential generalization of Pitman’s 2MX theorem. Electron. Commun. Probab. 7, 37–46 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    D.V. Choodnovsky, G.V. Choodnovsky, Pole expansions of nonlinear partial differential equations. Nuovo Cimento 40B, 339–353 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.V. Choodnovsky, G.V. Choodnovsky, Many body systems with 1∕r 2 potential, the Riccati equation, and completely integrable systems connected with the Schrödinger equation. Lett. Nuovo Cimento 23, 503–508 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Corwin, A. Hammond, KPZ line ensemble [arXiv:1312.2600]Google Scholar
  5. 5.
    E.B. Dynkin, Markov Processes, vol. 1 (Springer, Berlin, 1965)zbMATHCrossRefGoogle Scholar
  6. 6.
    S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986)zbMATHCrossRefGoogle Scholar
  7. 7.
    P.J. Forrester, Log-Gases and Random Matrices (Princeton University Press, Princeton, 2012)Google Scholar
  8. 8.
    A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin, On a Gauss-Givental representation of quantum Toda chain wave equation. Int. Math. Res. Not. 1–23 (2006)Google Scholar
  9. 9.
    M. Hallnäs, S.N.M. Ruijsenaars, Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and Toda systems. J. Math. Phys. 53, 123512 (2012)CrossRefGoogle Scholar
  10. 10.
    M. Hallnäs, S.N.M. Ruijsenaars, A recursive construction of joint eigenfunctions for the hyperbolic nonrelativistic Calogero-Moser Hamiltonians International Mathematics Research Notices, doi:10.1093/imrn/rnu267 (2015)zbMATHGoogle Scholar
  11. 11.
    M. Kac, P. Van Moerbeke, On some periodic Toda lattices. Proc. Natl. Acad. Sci. 72, 1627–1629 (1975)zbMATHCrossRefGoogle Scholar
  12. 12.
    F.P. Kelly, Markovian functions of a Markov chain. Sankya Ser. A 44, 372–379 (1982)zbMATHGoogle Scholar
  13. 13.
    J.G. Kemeny, J.L. Snell, Finite Markov Chains (Van Nostrand, Princeton, 1960)zbMATHGoogle Scholar
  14. 14.
    T.G. Kurtz, Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3, 1–29 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    T.G. Kurtz, Equivalence of stochastic equations and martingale problems, in Stochastic Analysis 2010, ed. by D. Crisan (Springer, New York, 2011)Google Scholar
  16. 16.
    V.B. Kuznetsov, E.K. Sklyanin, On Bäcklund transformations for many-body systems. J. Phys. A Math. Gen. 31, 2241–2251 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Matsumoto, M. Yor, A version of Pitman’s 2MX theorem for geometric Brownian motions. C. R. Acad. Sci. Paris 328, 1067–1074 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    G. Moreno Flores, J. Quastel, D. Remenik, In preparationGoogle Scholar
  19. 19.
    NIST Digital Library of Mathematical Functions (2013)., Release 1.0.6 of 2013-05-06 [Online companion to [20]]
  20. 20.
    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010) [Print companion to [19]]Google Scholar
  21. 21.
    N. O’Connell, Directed polymers and the quantum Toda lattice. Ann. Probab. 40, 437–458 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    N. O’Connell, Geometric RSK and the Toda lattice. Illinois Journal of Mathematics, 57(3), 883–918 (2013)zbMATHMathSciNetGoogle Scholar
  23. 23.
    N. O’Connell, J. Warren, A multi-layer extension of the stochastic heat equation [arXiv:1104.3509]Google Scholar
  24. 24.
    V. Pasquier, M. Gaudin, The periodic Toda chain and a matrix generalization of the Bessel function recursion relations. J. Phys. A Math. Gen. 25, 5243–5252 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    J.W. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab. 7, 511–526 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    L.C.G. Rogers, J.W. Pitman, Markov functions. Ann. Probab. 9, 573–582 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    E.K. Sklyanin, Bäcklund transformations and Baxter’s Q-operator, in Integrable Systems: From Classical to Quantum, CRM Proceedings, vol. 26 (2000), pp. 227–250Google Scholar
  28. 28.
    M. Toda, Theory of Nonlinear Lattices (Springer, Berlin, 1981)zbMATHCrossRefGoogle Scholar
  29. 29.
    S. Wojciechowski, The analogue of the Bäcklund transformation for integrable many-body systems. J. Phys. A Math. Gen. 15, L653–L657 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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