The Continuum Directed Random Polymer

Abstract

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter β, and for a given β and realization of the noise the path is a Markov process. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all β>0 and for almost all realizations of the white noise the path measure has the same Hölder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).

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References

  1. 1.

    Alberts, T., Khanin, K., Quastel, J.: Intermediate disorder regime for directed polymers in dimension 1+1. Phys. Rev. Lett. 105(9), 090603 (2010)

    ADS  Article  Google Scholar 

  2. 2.

    Alberts, T., Khanin, K., Quastel, J.: Intermediate disorder regime for 1+1 dimensional directed polymers (2012). arXiv:1202.4398

  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. (2010). doi:10.1002/cpa.20347

    Google Scholar 

  4. 4.

    Bertini, L., Cancrini, N.: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78(5–6), 1377–1401 (1995)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Bezerra, S., Tindel, S., Viens, F.: Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36(5), 1642–1675 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Bolthausen, E.: A note on the diffusion of directed polymers in a random environment. Commun. Math. Phys. 123(4), 529–534 (1989)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Comets, F., Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Comets, F., Shiga, T., Yoshida, N.: Probabilistic analysis of directed polymers in a random environment: a review. In: Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math., vol. 39, pp. 115–142. Math. Soc. Japan, Tokyo (2004)

    Google Scholar 

  10. 10.

    Conlon, J.G., Olsen, P.A.: A Brownian motion version of the directed polymer problem. J. Stat. Phys. 84(3–4), 415–454 (1996)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  12. 12.

    Giambattista, G.: Random Polymer Models. Imperial College Press, London (2007)

    MATH  Google Scholar 

  13. 13.

    Huse, D.A., Henley, C.L.: Pinning and roughening of domain walls in ising systems due to random impurities. Phys. Rev. Lett. 54(25), 2708–2711 (1985)

    ADS  Article  Google Scholar 

  14. 14.

    Imbrie, J.Z., Spencer, T.: Diffusion of directed polymers in a random environment. J. Stat. Phys. 52(3–4), 609–626 (1988)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  16. 16.

    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)

    ADS  Article  MATH  Google Scholar 

  17. 17.

    Kifer, Y.: The Burgers equation with a random force and a general model for directed polymers in random environments. Probab. Theory Relat. Fields 108(1), 29–65 (1997)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Mueller, C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Varadhan, S.R.S.: Stochastic Processes. Courant Lecture Notes in Mathematics, vol. 16. Courant Institute of Mathematical Sciences, New York (2007)

    MATH  Google Scholar 

  20. 20.

    Walsh, J.B.: An introduction to stochastic partial differential equations. In: École D’été de Probabilités de Saint–Flour, XIV—1984. Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)

    Google Scholar 

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Correspondence to Jeremy Quastel.

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Research of all three authors supported by the Natural Sciences and Engineering Research Council of Canada.

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Alberts, T., Khanin, K. & Quastel, J. The Continuum Directed Random Polymer. J Stat Phys 154, 305–326 (2014). https://doi.org/10.1007/s10955-013-0872-z

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Keywords

  • Directed random polymers
  • KPZ