Self-Duality of Markov Processes and Intertwining Functions

  • Chiara FranceschiniEmail author
  • Cristian Giardinà
  • Wolter Groenevelt


We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitarily equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.


Stochastic duality Representation theory Lie algebras Orthogonal polynomials 

Mathematics Subject Classification (2010)

60J25 60J27 60J60 82C05 


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encycl. Math. Appl. 71 Cambridge University Press (1999)Google Scholar
  2. 2.
    Belitsky, V., Schütz, G.M.: Self-duality for the two-component asymmetric simple exclusion process. J. Math. Phys. 56(8), 083302 (2015)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Belitsky, V., Schütz, G.M.: Quantum algebra symmetry of the ASEP with second-class particles. J. Stat. Phys. 161(5), 821–842 (2015)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Belitsky, V., Schütz, G.M.: Self-duality and shock dynamics in the n-species priority ASEP. Stochastic Processes and their Applications 128(4), 1165–1207 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bernardin, C.: Superdiffusivity of asymmetric energy model in dimensions 1 and 2. J. Math. Phys. 49(10), 103301 (2008)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Borodin, A.: On a family of symmetric rational functions. Adv. Math. 306, 973–1018 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke Math. J. 165(3), 563–624 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carinci, G., Giardinà, C., Giberti, C., Redig, F.: Duality for stochastic models of transport. J. Stat. Phys. 152(4), 657–697 (2013)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Carinci, G., Giardin, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with \(U_q(\mathfrak {sl}_2)\) stochastic duality. Probab. Theory Relat. Fields 166, 887–933 (2016)CrossRefGoogle Scholar
  11. 11.
    Carinci, G., Giardinà, C., Redig, F., Sasamoto, T.: Asymmetric Stochastic Transport Models with \({\mathscr{U}}_q (\mathfrak {su}(1, 1))\) Symmetry. J. Stat. Phys. 163(2), 239–279 (2016)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Carinci, G., Giardinà, C., Giberti, C., Redig, F.: Dualities in population genetics: a fresh look with new dualities. Stochastic Processes and their Applications 125(3), 941–969 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, Z., de Gier, J., Wheeler, M.: Integrable stochastic dualities and the deformed Knizhnik-Zamolodchikov equation. Preprint arXiv:1709.06227. International Mathematics Research Notices, rny159 (2017)
  14. 14.
    Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Corwin, I., Shen, H., Tsai, L.-C.: ASEP(q, j) converges to the KPZ equation (2016)Google Scholar
  16. 16.
    De Masi, A., Presutti, E.: Mathematical methods for hydrodynamic limits. Springer (2006)Google Scholar
  17. 17.
    Franceschini, C., Giardinà, C.: Stochastic Duality and Orthogonal Polynomials. arXiv:1701.09115 (2017)
  18. 18.
    Giardinà, C., Kurchan, J.: The Fourier law in a momentum-conserving chain. J. Stat. Mech: Theory Exp. 05(2005), P05009 (2005)Google Scholar
  19. 19.
    Giardinà, C., Kurchan, J., Redig, F.: Duality and exact correlations for a model of heat conduction. J. Math. Phys. 48(3), 033301 (2007)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Giardinà, C., Kurchan, J., Redig, F., Vafayi, K.: Duality and hidden symmetries in interacting particle systems. J. Stat. Phys. 135(1), 25–55 (2009)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Giardinà, C., Redig, F., Vafayi, K.: Correlation inequalities for interacting particle systems with duality. J. Stat. Phys. 141(2), 242–263 (2010)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Groenevelt, W.: Orthogonal stochastic duality functions from Lie algebra representations. arXiv:1709.05997 (2017)
  23. 23.
    Jansen, S., Kurt, N.: On the notion(s) of duality for Markov processes. Probab. Surv. 11, 59–120 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Keisling, J.D.: An ergodic theorem for the symmetric generalized exclusion process. Markov Process. Relat. Fields 4, 351–379 (1998)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27(1), 65–74 (1982)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer (2010)Google Scholar
  27. 27.
    Koelink, H.T., Van der Jeugt, J.: Convolution for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Annal. 29, 794–822 (1998)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kuan, J.: Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two. J. Phys. A Math. Theor. 49(11), 115002 (2016)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kuan, J.: A Multi-species ASEP(q, j) and q-TAZRP with stochastic duality. arXiv:1605.00691 (2016)
  30. 30.
    Kuan, J.: An algebraic construction of duality functions for the stochastic \( {U_q} (A_n^{(1)})\) vertex model and its degenerations. arXiv:1701.04468 (2017)
  31. 31.
    Liggett, T.M.: Interacting particles systems. Springer (1985)Google Scholar
  32. 32.
    Mangazeev, V.: On the Yang–Baxter equation for the six-vertex model. Nucl. Phys. B 882, 70–96 (2014)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Möhle, M.: The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5(5), 761–777 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Redig, F., Sau, F.: Duality functions and stationary product measures. arXiv:1702.07237 (2017)
  35. 35.
    Sasamoto, T., Spohn, H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104(23), 230602 (2010)ADSCrossRefGoogle Scholar
  36. 36.
    Schütz, G.M.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86(5), 1265–1287 (1997)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Schütz, G.M., Sandow, S.: Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems. Phys. Rev. E 49(4), 2726 (1994)ADSCrossRefGoogle Scholar
  38. 38.
    Spitzer, F.: Interaction of Markov processes. Advances in Mathematics 5(2), 246–290 (1970)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Spohn, H.: Long range correlations for stochastic lattice gases in a non-equilibrium steady state. J. Phys. A Math. Gen. 16(18), 4275 (1983)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.University of FerraraFerraraItaly
  2. 2.University of Modena and Reggio EmiliaModenaItaly
  3. 3.Technische Universiteit Delft, DIAMDelftThe Netherlands

Personalised recommendations