Advertisement

Acta Applicandae Mathematicae

, Volume 151, Issue 1, pp 81–88 | Cite as

Stochastic Quantization for the Fractional Edwards Measure

  • Wolfgang Bock
  • Torben Fattler
  • Ludwig Streit
Article
  • 131 Downloads

Abstract

We prove that there exists a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion, \(\mu_{ {g,H}}\), \(H\in (0,1)\) for \(dH < 1\). The diffusion is constructed in the framework of Dirichlet forms in infinite dimensional (Gaussian) analysis. Moreover, the process is invariant under time translations.

Keywords

Stochastic quantization Edwards model Fractional Brownian motion Dirichlet forms White noise analysis 

Notes

Acknowledgements

We truly thank M. Röckner for helpful discussions. Furthermore we thank M. Grothaus and M. J. Oliveira for helpful comments. Moreover, the authors are grateful for the referee’s constructive comments. Financial support by CRC 701 and the mathematics department of the University of Kaiserslautern for research visits at Bielefeld university are gratefully acknowledged.

References

  1. 1.
    Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions, solutions via Dirichlet forms. Probab. Theory Relat. Fields 89, 347–386 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Hu, Y.-Z., Röckner, M., Zhou, X.Y.: Stochastic quantization of the two-dimensional polymer measure. Appl. Math. Optim. 40, 341–354 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bass, R.F., Khoshnevisan, D.: Intersection local times and Tanaka formulas. Ann. Inst. Henri Poincaré 29, 419–451 (1993) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berezansky, Yu.M., Kondratiev, Yu.G.: Spectral Methods in Infinite-Dimensional Analysis. Naukova Dumka, Kiev (1988) (in Russian). English translation, Kluwer Academic Publishers, Dordrecht, 1995 Google Scholar
  5. 5.
    Bock, W., Fattler, T., Streit, L.: Stochastic Quantization of the fractional Edwards measure in the case \(Hd=1\), in preparation Google Scholar
  6. 6.
    Bock, W., Fattler, T., Streit, L., Tse, O.: Discrete fractional polymer measures: stochastic quantization and numerics, in preparation Google Scholar
  7. 7.
    de Faria, M., Drumond, C., Streit, L.: The renormalization of self-intersection local times. I. The chaos expansion. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3(2), 223–236 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dvoretzky, A., Erdös, P., Kakutani, S.: Double points of paths of Brownian motion in n-space. Acta Sci. Math. 12, 75–81 (1950) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dvoretzky, A., Erdös, P., Kakutani, S., Taylor, S.J.: Triple points of the Brownian motion in 3-space. Proc. Camb. Philos. Soc. 53, 856–862 (1957) CrossRefzbMATHGoogle Scholar
  10. 10.
    Fukushima, M., Oshima, Y., Takeda, T.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin (1994) CrossRefzbMATHGoogle Scholar
  11. 11.
    He, S.W., Yang, W.Q., Yao, R.Q., Wang, J.G.: Local times of self-intersection for multidimensional Brownian motion. Nagoya Math. J. 138, 51–67 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hida, T.: Stationary Stochastic Processes. Mathematical Notes Series. Princeton University Press, Princeton (1970) zbMATHGoogle Scholar
  13. 13.
    Hida, T., Kuo, H.H., Potthoff, J., Streit, L.: White Noise. an Infinite Dimensional Calculus. Kluwer, Dordrecht (1993), 516 pp. zbMATHGoogle Scholar
  14. 14.
    Hu, Y.: Self-intersection local time of fractional Brownian motions—via chaos expansion. J. Math. Kyoto Univ. 41, 233–250 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hu, Y., Kallianpur, G.: Exponential integrability and application to stochastic quantization. Appl. Math. Optim. 37, 295–353 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33(3), 948–983 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hu, Y., Nualart, D.: Regularity of renormalized self-intersection local time for fractional Brownian motion. Commun. Inf. Syst. 7, 21–30 (2007) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hu, Y., Nualart, D., Song, J.: Integral representation of renormalized self-intersection local times. J. Funct. Anal. 255(9), 2507–2532 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hu, Y., Øksendal, B.: Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 06(1), 1–32 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Imkeller, P., Pérez-Abreu, V., Vives, J.: Chaos expansions of double intersection local times of Brownian motion in \(\mathbb{R}^{d}\) and renormalization. Stoch. Process. Appl. 56, 1–34 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976) zbMATHGoogle Scholar
  22. 22.
    Kondratiev, Yu.G., Streit, L., Westerkamp, W.: A note on positive distributions in Gaussian analysis. Ukr. Math. J. 47(5), 749–759 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Le Gall, J.F.: Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. In: Sém. Prob. XIX. Lecture Notes in Mathematics, vol. 1123, pp. 314–331. Springer, Berlin (1985) Google Scholar
  24. 24.
    Lévy, P.: Le mouvement brownien plan. Am. J. Math. 62, 487–550 (1940) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lyons, T.J.: The critical dimension at which quasi-every Brownian motion is self-avoiding. Adv. Appl. Probab. Spec. Suppl., 87–99 (1986) zbMATHGoogle Scholar
  26. 26.
    Ma, Z.-M., Röckner, M.: An Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992) CrossRefzbMATHGoogle Scholar
  27. 27.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Probability and Its Applications. Springer, Berlin (2006) zbMATHGoogle Scholar
  28. 28.
    Obata, N.: White Noise Calculus and Fock Spaces. LNM, vol. 1577. Springer, Berlin (1994) CrossRefzbMATHGoogle Scholar
  29. 29.
    Parisi, G., Wu, Y.-S.: Perturbation theory without gauge fixing. Sci. Sin. 24, 483 (1981) MathSciNetGoogle Scholar
  30. 30.
    Potthoff, J.: On differential operators in white noise analysis. Acta Appl. Math. 63, 333–347 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Symanzik, K.: Euclidean quantum field theory. In: Jost, R. (ed.) Local Quantum Theory. Academic Press, New York (1969) Google Scholar
  32. 32.
    Varadhan, S.R.S.: Appendix to “Euclidean quantum field theory” by K. Symanzik. In: Jost, R. (ed.) Local Quantum Theory. Academic Press, New York (1969) Google Scholar
  33. 33.
    Watanabe, H.: The local time of self-intersections of Brownian motions as generalized Brownian functionals. Lett. Math. Phys. 23, 1–9 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Westwater, J.: On Edward’s model for long polymer chains. Commun. Math. Phys. 72, 131–174 (1980) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wolpert, R.: Wiener path intersection and local time. J. Funct. Anal. 30, 329–340 (1978) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yor, M.: Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement brownien dans \({\mathbb{R}}^{3}\). In: Séminaire de Probabilité. Lecture Notes in Mathematics, vol. 1123, pp. 350–365. Springer, Berlin (1985) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Technomathematics GroupUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Functional Analysis and Stochastic Analysis GroupUniversity of KaiserslauternKaiserslauternGermany
  3. 3.BIBOSBielefeldGermany
  4. 4.CIMA-UMAFunchalPortugal

Personalised recommendations