Advertisement

Communications in Mathematical Physics

, Volume 341, Issue 1, pp 1–33 | Cite as

A Multi-Layer Extension of the Stochastic Heat Equation

Open Access
Article

Abstract

Motivated by recent developments on solvable directed polymer models, we define a ‘multi-layer’ extension of the stochastic heat equation involving non-intersecting Brownian motions. By developing a connection with Darboux transformations and the two-dimensional Toda equations, we conjecture a Markovian evolution in time for this multi-layer process. As a first step in this direction, we establish an analogue of the Karlin-McGregor formula for the stochastic heat equation and use it to prove a special case of this conjecture.

Keywords

Markov Property Darboux Transformation Brownian Bridge Asymmetric Simple Exclusion Process Directed Polymer 
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.

References

  1. 1.
    Alberts, T., Khanin, K., Quastel, J.: The intermediate disorder regime for directed polymers in dimension 1+1. Ann. Probab. 42(3), 1212–1256 (2014)Google Scholar
  2. 2.
    Alberts T., Khanin K., Quastel J.: The continuum directed random polymer. J. Stat. Phys. 154, 305–326 (2014)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math. 64, 466–537 (2011)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Arrigo DJ., Hickling F.: Darboux transformations and linear parabolic partial differential equations. J. Phys. A 35, L389–L399 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Arrigo DJ., Hickling F.: An nth-order Darboux transformation for the one-dimensional time-dependent Schrödinger equation. J. Phys. A 36, 1615–1621 (2003)MATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Balazs M., Quastel J., Seppalainen T.: Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers. J. Am. Math. Soc. 24, 683–708 (2011)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bertini L., Cancrini N.: The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78, 1377–1401 (1995)MATHMathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Biane Ph., Bougerol Ph., O’Connell N.: Continuous crystals and Duistermaat-Heckman measure for Coxeter groups. Adv. Math. 221, 1522–1583 (2009)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Borodin A., Corwin I.: Macdonald processes. Probab. Theor. Rel. Fields 158, 225–400 (2014)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Calabrese P., Le Doussal P., Rosso A.: Free-energy distribution of the directed polymer at high temperature. EPL 90, 20002 (2010)CrossRefADSGoogle Scholar
  12. 12.
    Cépa E., Lépingle D.: Diffusing particles with electrostatic repulsion. Probab. Theory Relat. Fields 107, 429–449 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cohn D.L.: Measurable choice of limit points and the Existence of Separable and Measurable Processes. Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 161–165 (1972)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Corwin, I., Hammond, A.: KPZ line ensemble. Probab. Theor. Rel. Fields (2015) (online)Google Scholar
  15. 15.
    Corwin I., O’Connell N., Seppäläinen T., Zygouras N.: Tropical combinatorics and Whittaker functions. Duke Math. J. 163, 465–663 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dotsenko, V.: Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers. J. Stat. Mech. P07010 (2010)Google Scholar
  17. 17.
    Dotsenko V.: Bethe ansatz derivation of the Tracy-Widom distribution for one- dimensional directed polymers. EPL 90, 20003 (2010)CrossRefADSGoogle Scholar
  18. 18.
    Dotsenko, V., Klumov, B.: Bethe ansatz solution for one-dimensional directed polymers in random media. J. Stat. Mech. P03022 (2010)Google Scholar
  19. 19.
    Fitzsimmons, P., Pitman J., Yor, M.: Markovian bridges:construction, palm interpretation and splicing. In: Seminar on Stochastic prcocesses, pp. 101–134 (1992)Google Scholar
  20. 20.
    Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, New Jersey (1964)MATHGoogle Scholar
  21. 21.
    Hairer M.: Solving the KPZ equation. Ann. Math. 178, 559–664 (2013)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Harish-Chandra: Proc. Nat. Acad. Sci. 42, 252 (1956)Google Scholar
  23. 23.
    Horn R., Johnson C.: Matrix Analysis. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  24. 24.
    Holden, H., Oksendal, B., Uboe, J., Zhang, T.S.: Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, 2nd edn. Universitext, Springer Verlag, Heidelberg (2010)Google Scholar
  25. 25.
    Itzykson C., Zuber J.-B.: J. Math. Phys. 21, 411 (1980)MATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Johansson, K.: Random matrices and determinantal processes. In: Mathematical Statistical Physics, pp. 1–55. Elsevier, Amsterdam (2006)Google Scholar
  27. 27.
    Jones L., O’Connell N.: Weyl chambers, symmetric spaces and number variance saturation. ALEA 2, 91–118 (2006)MATHMathSciNetGoogle Scholar
  28. 28.
    Kardar M., Parisi G., Zhang Y.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)MATHCrossRefADSGoogle Scholar
  29. 29.
    Karlin S., McGregor J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Kirillov, A.N.: Introduction to tropical combinatorics. In: Kirillov, A.N., Liskova. N. (eds.) Physics and Combinatorics. Proc. Nagoya 2000, 2nd Internat, pp. 82–150. Workshop World Scientific, Singapore (2001)Google Scholar
  31. 31.
    Moreno-Flores, G., Quastel, J., Remenik, D.: Intermediate disorder limits for directed polymers with boundary conditions. In preparationGoogle Scholar
  32. 32.
    Moriarty J., O’Connell N.: On the free energy of a directed polymer in a Brownian environment. Markov Process. Related Fields 13, 251–266 (2007)MATHMathSciNetGoogle Scholar
  33. 33.
    Noumi, M., Yamada, Y.: Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions. Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, pp. 371–442, Math. Soc. Japan, Tokyo (2004)Google Scholar
  34. 34.
    O’Connell N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40, 437–458 (2012)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    O’Connell N., Yor M.: Brownian analogues of Burke’s theorem. Stoch. Process. Appl. 96, 285–304 (2001)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    O’Connell N., Seppäläinen T., Zygouras N.: Geometric RSK correspondence, Whittaker functions and symmetrized random polymers. Inventiones Math. 197, 361–416 (2014)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Pitman J.: The distribution of local times of a Brownian bridge. Séminaires de Probabilités 33, 388–394 (1999)MathSciNetMATHGoogle Scholar
  38. 38.
    Prähofer M., Spohn H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Prolhac, S., Spohn, H.: Two-point generating function of the free energy for a directed polymer in a random medium. J. Stat. Mech. P01031 (2011)Google Scholar
  40. 40.
    Prolhac, S., Spohn, H.: The one-dimensional KPZ equation and the Airy process. J. Stat. Mech. P03020 (2011)Google Scholar
  41. 41.
    Quastel, J.: Introduction to KPZ. Current Developments in Mathematics. International Press, Boston (2011)Google Scholar
  42. 42.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin (1999)MATHCrossRefGoogle Scholar
  43. 43.
    Sasamoto T., Spohn H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834, 523–542 (2010)MATHMathSciNetCrossRefADSGoogle Scholar
  44. 44.
    Sasamoto T., Spohn H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010)CrossRefADSGoogle Scholar
  45. 45.
    Sasamoto T., Spohn H.: The crossover regime for the weakly asymmetric simple exclusion process. J. Stat. Phys. 140, 209–231 (2010)MATHMathSciNetCrossRefADSGoogle Scholar
  46. 46.
    Sasamoto, T., Spohn, H.: The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class. J. Stat. Mech. P11013 (2010)Google Scholar
  47. 47.
    Seppäläinen T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40, 19–73 (2012)MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Seppäläinen T., Valkó B.: Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment. Alea 7, 451–476 (2010)MATHMathSciNetGoogle Scholar
  49. 49.
    Tracy, C.A., Widom. H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008); Erratum: Commun. Math. Phys. 304, 875–878 (2011)Google Scholar
  50. 50.
    Tracy C.A., Widom H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291–300 (2008)MATHMathSciNetCrossRefADSGoogle Scholar
  51. 51.
    Tracy C.A., Widom H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009)MATHMathSciNetCrossRefADSGoogle Scholar
  52. 52.
    Tracy C.A., Widom H.: Formulas for joint probabilities for the asymmetric simple exclusion process. J. Math. Phys. 51, 063302 (2010)MathSciNetCrossRefMATHADSGoogle Scholar
  53. 53.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: Ecole d’Eté de Probabilités de Saint Flour XIV, Lecture Notes in Mathematics, vol. 1180, pp. 265–438, Springer-Verlag, Berlin (1986)Google Scholar
  54. 54.
    Wenchang C.: Finite differences and determinant identities. Lin. Alg. Appl. 430, 215–228 (2009)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of StatisticsUniversity of WarwickCoventryUK

Personalised recommendations