Stochastic Bäcklund Transformations

Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

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.

Keywords

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.

Notes

Acknowledgements

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.

References

  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)MATHCrossRefGoogle Scholar
  6. 6.
    S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986)MATHCrossRefGoogle 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)MATHGoogle Scholar
  11. 11.
    M. Kac, P. Van Moerbeke, On some periodic Toda lattices. Proc. Natl. Acad. Sci. 72, 1627–1629 (1975)MATHCrossRefGoogle Scholar
  12. 12.
    F.P. Kelly, Markovian functions of a Markov chain. Sankya Ser. A 44, 372–379 (1982)MATHGoogle Scholar
  13. 13.
    J.G. Kemeny, J.L. Snell, Finite Markov Chains (Van Nostrand, Princeton, 1960)MATHGoogle Scholar
  14. 14.
    T.G. Kurtz, Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3, 1–29 (1998)MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    G. Moreno Flores, J. Quastel, D. Remenik, In preparationGoogle Scholar
  19. 19.
    NIST Digital Library of Mathematical Functions (2013). http://dlmf.nist.gov/, 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)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    N. O’Connell, Geometric RSK and the Toda lattice. Illinois Journal of Mathematics, 57(3), 883–918 (2013)MATHMathSciNetGoogle 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)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    J.W. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab. 7, 511–526 (1975)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    L.C.G. Rogers, J.W. Pitman, Markov functions. Ann. Probab. 9, 573–582 (1981)MATHMathSciNetCrossRefGoogle 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)MATHCrossRefGoogle 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

Personalised recommendations