# Stochastic Bäcklund Transformations

Chapter

## 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.F. Baudoin, Further exponential generalization of Pitman’s 2
*M*−*X*theorem. Electron. Commun. Probab.**7**, 37–46 (2002)MathSciNetCrossRefGoogle Scholar - 2.D.V. Choodnovsky, G.V. Choodnovsky, Pole expansions of nonlinear partial differential equations. Nuovo Cimento
**40B**, 339–353 (1977)MathSciNetCrossRefGoogle Scholar - 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.I. Corwin, A. Hammond, KPZ line ensemble [arXiv:1312.2600]Google Scholar
- 5.E.B. Dynkin,
*Markov Processes*, vol. 1 (Springer, Berlin, 1965)MATHCrossRefGoogle Scholar - 6.S.N. Ethier, T.G. Kurtz,
*Markov Processes: Characterization and Convergence*(Wiley, New York, 1986)MATHCrossRefGoogle Scholar - 7.P.J. Forrester,
*Log-Gases and Random Matrices*(Princeton University Press, Princeton, 2012)Google Scholar - 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.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.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.M. Kac, P. Van Moerbeke, On some periodic Toda lattices. Proc. Natl. Acad. Sci.
**72**, 1627–1629 (1975)MATHCrossRefGoogle Scholar - 12.F.P. Kelly, Markovian functions of a Markov chain. Sankya Ser. A
**44**, 372–379 (1982)MATHGoogle Scholar - 13.J.G. Kemeny, J.L. Snell,
*Finite Markov Chains*(Van Nostrand, Princeton, 1960)MATHGoogle Scholar - 14.T.G. Kurtz, Martingale problems for conditional distributions of Markov processes. Electron. J. Probab.
**3**, 1–29 (1998)MATHMathSciNetCrossRefGoogle Scholar - 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.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.H. Matsumoto, M. Yor, A version of Pitman’s 2
*M*−*X*theorem for geometric Brownian motions. C. R. Acad. Sci. Paris**328**, 1067–1074 (1999)MATHMathSciNetCrossRefGoogle Scholar - 18.G. Moreno Flores, J. Quastel, D. Remenik, In preparationGoogle Scholar
- 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.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.N. O’Connell, Directed polymers and the quantum Toda lattice. Ann. Probab.
**40**, 437–458 (2012)MATHMathSciNetCrossRefGoogle Scholar - 22.N. O’Connell, Geometric RSK and the Toda lattice. Illinois Journal of Mathematics,
**57**(3), 883–918 (2013)MATHMathSciNetGoogle Scholar - 23.N. O’Connell, J. Warren, A multi-layer extension of the stochastic heat equation [arXiv:1104.3509]Google Scholar
- 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.J.W. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab.
**7**, 511–526 (1975)MATHMathSciNetCrossRefGoogle Scholar - 26.L.C.G. Rogers, J.W. Pitman, Markov functions. Ann. Probab.
**9**, 573–582 (1981)MATHMathSciNetCrossRefGoogle Scholar - 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.M. Toda,
*Theory of Nonlinear Lattices*(Springer, Berlin, 1981)MATHCrossRefGoogle Scholar - 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

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