Skip to main content
SpringerLink
Log in

The Parabolic Harnack Inequality for the Time Dependent Ginzburg–Landau Type SPDE and its Application

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg–Landau type stochastic partial differential equation (=SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P(φ)1-time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry–Emery’s Γ2-method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aida, S. and Kawabi, H.: ‘Short time asymptotics of a certain infinite dimensional diffusion process’, in L. Decreusefond, B. Øksendal and A.S. Üstünel (eds), Stochastic Analysis and Related Topics VII: Proceedings of 7-th Silvri Workshop, Progress in Probability 48, Birkhäuser, 2001, pp. 77–124.

  2. Aida, S. and Stroock, D.: ‘Moment estimates derived from Poincaré and logarithmic Sobolev inequalities’, Math. Res. Lett. 1 (1994), 75–86.

    Google Scholar 

  3. Aida, S. and Zhang, T.S.: ‘On the small time asymptotics of diffusion processes on path groups’, Potential Anal. 16 (2002), 67–78.

    Google Scholar 

  4. Albeverio, S. and Kusuoka, S.: ‘Maximality of infinite-dimensional Dirichlet forms and Høegh-Krohn’s model of quantum fields’, in Ideas and Methods in Quantum and Statistical Physics (Oslo 1988), Cambridge Univ. Press, Cambridge, 1992, pp. 301–330.

    Google Scholar 

  5. Albeverio, S. and Röckner, M.: ‘Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms’, Probab. Theory Related Fields 89 (1991), 347–386.

    Google Scholar 

  6. Bakry, D.: ‘On Sobolev and logarithmic Sobolev inequalities for Markov semigroups’, in K.D. Elworthy, S. Kusuoka and I. Shigekawa (eds), New Trends in Stochastic Analysis, World Sci. Publishing, River Edge, NJ, 1997, pp. 43–75.

    Google Scholar 

  7. Bakry, D. and Emery, M.: ‘Diffusions hypercontractives’, in Séminaire de probabilités XIX, Lecture Notes in Math. 1123, Springer, 1985, pp. 177–206.

  8. Fang, S.: ‘On the Ornstein–Uhlenbeck process’, Stochastics Stochastic Rep. 46 (1994), 141–159.

    Google Scholar 

  9. Funaki, T.: ‘Random motion of strings and related stochastic evolution equations’, Nagoya Math. J. 89 (1983), 129–193.

    Google Scholar 

  10. Funaki, T.: ‘The reversible measure of multi-dimensional Ginzburg–Landau continuum model’, Osaka J. Math. 28 (1991), 463–494.

    Google Scholar 

  11. Funaki, T.: ‘SPDE approach and log-Sobolev inequalities for continuum field with two-body interactions’, Preprint, 1996. To appear in Proceedings of Japan-German Seminar, 1996, Hiroshima, World Sci.

  12. Hino, M.: ‘On short time asymptotic behavior of some symmetric diffusions on general state spaces’, Potential Anal. 16 (2002), 249–264.

    Google Scholar 

  13. Hino, M. and Ramirez, J.A.: ‘Small time Gaussian behavior of symmetric diffusion semigroups’, Ann. Probab. 31 (2003), 1254–1295.

    Google Scholar 

  14. Iwata, K.: ‘Reversible measures of a P(φ)1-time evolution’, in K. Itô and N. Ikeda (eds), Prob. Meth. in Math. Phys.: Proceedings of Taniguchi Symposium, Kinokuniya, 1985, pp. 195–209.

  15. Iwata, K.: ‘An infinite dimensional stochastic differential equation with state space C(R)’, Probab. Theory Related Fields 74 (1987), 141–159.

    Google Scholar 

  16. Kusuoka, S.: ‘Dirichlet forms and diffusion processes on Banach space’, J. Fac. Sci. Univ. Tokyo Sec. IA 29 (1982), 79–95.

    Google Scholar 

  17. Kusuoka, S.: ‘The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity (I)’, J. Fac. Sci. Univ. Tokyo Sec. IA 29 (1982), 567–598.

    Google Scholar 

  18. Ma, Z.-M. and Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992.

    Google Scholar 

  19. Øksendal, B.: Stochastic Differential Equations, An Introduction with Applications, 5th edn, Springer-Verlag, Berlin, 1998.

    Google Scholar 

  20. Ramirez, J.A.: ‘Short-time asymptotics in Dirichlet spaces’, Comm. Pure Appl. Math. 54 (2001), 259–293.

    Google Scholar 

  21. Röckner, M. and Wang, F.-Y.: ‘Harnack and functional inequalities for generalized Mehler semigroups’, J. Funct. Anal. 203 (2003), 237–261.

    Google Scholar 

  22. Wang, F.-Y.: ‘Logarithmic Sobolev inequalities on noncompact Riemannian manifolds’, Probab. Theory Related Fields 109 (1997), 417–424.

    Google Scholar 

  23. Zhang, T.-S.: ‘On the small time asymptotics of diffusion processes on Hilbert spaces’, Ann. Probab. 28 (2000), 537–557.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Hiroshi Kawabi

Authors
  1. Hiroshi Kawabi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroshi Kawabi.

Additional information

Mathematics Subject Classifications (2000)

Primary: 60H15, 47D07; secondary: 60J60, 31C25.

Hiroshi Kawabi: The author was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists and by Grant-in-Aid for Scientific research 15-03706, Japan Society for the Promotion of Science, Japan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kawabi, H. The Parabolic Harnack Inequality for the Time Dependent Ginzburg–Landau Type SPDE and its Application. Potential Anal 22, 61–84 (2005). https://doi.org/10.1007/s11118-003-6456-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-003-6456-9

Keywords

Search

Navigation