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Persistence of Gaussian processes: non-summable correlations

Abstract

Suppose the auto-correlations of real-valued, centered Gaussian process \(Z(\cdot )\) are non-negative and decay as \(\rho (|s-t|)\) for some \(\rho (\cdot )\) regularly varying at infinity of order \(-\alpha \in [-1,0)\). With \(I_\rho (t)=\int _0^t \rho (s)ds\) its primitive, we show that the persistence probabilities decay rate of \( -\log \mathbb {P}(\sup _{t \in [0,T]}\{Z(t)\}<0)\) is precisely of order \((T/I_\rho (T)) \log I_\rho (T)\), thereby closing the gap between the lower and upper bounds of Newell and Rosenblatt (Ann. Math. Stat. 33:1306–1313, 1962), which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of Sakagawa (Adv. Appl. Probab. 47:146–163, 2015) about the dependence on d of such persistence decay for the Langevin dynamics of certain \(\nabla \phi \)-interface models on \(\mathbb {Z}^d\).

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References

  1. Adler, R.J., Taylor, J.E.: Random fields and geometry. Springer, New York (2007)

    MATH  Google Scholar 

  2. Aurzada, F., Simon, T.: Persistence probabilities and exponents. Lévy matters V, Lecture Notes in Math., vol. 2149, pp. 183–221. Springer, New York (2015)

  3. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  4. Bryc, W.L., Dembo, A.: On large deviations of empirical measures for stationary Gaussian processes. Stoch. Proc. Appl. 58, 23–34 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bray, A.J., Majumdar, S.N., Schehr, G.: Persistence and first-passage properties in non-equilibrium systems. Adv. Phys. 62(3), 225–361 (2013)

    Article  Google Scholar 

  6. Dembo, A., Deuschel, J.D.: Aging for interacting diffusion processes. Ann. Inst. H. Poincare Probab. Statist. 43, 461–480 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Deuschel, J.D.: Invariance principle and empirical mean large deviations of the critical Ornstein-Uhlenbeck process. Ann. Prob. 17, 74–90 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Deuschel, J.D.: The random walk representation for interacting diffusion processes, Interacting stochastic systems, 377–393. Springer, Berlin (2005)

    Google Scholar 

  9. Dembo, A., Mukherjee, S.: No zero-crossings for random polynomials and the heat equation. Ann. Probab. 43(1), 85–118 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dembo, A., Poonen, B., Shao, Q.M., Zeitouni, O.: Random polynomials having few or no real zeros. J. Am. Math. Soc. 15(4), 857–892 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Donsker, M.D., Varadhan, S.R.S.: Large deviations for stationary Gaussian processes. Commun. Math. Phys. 97(1–2), 187–210 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Feldheim, N., Feldheim, O.: Long gaps between sign-changes of Gaussian stationary processes. Int. Math. Res. Notices 11, 3021–3034 (2015)

    MATH  MathSciNet  Google Scholar 

  13. Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg-Landau interface model. Commun. Math. Phys. 185(1), 1–36 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Garet, O.: Infinite dimensional dynamics associated to quadratic Hamiltonians. Markov Process. Related Fields 6, 205–237 (2000)

    MATH  MathSciNet  Google Scholar 

  15. Giacomin, G., Olla, S., Spohn, H.: Equilibrium fluctuations for \( \nabla \phi \) interface model. Ann. Probab. 29(3), 1138–1172 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gambassi, A., Paul, R., Schehr, G.: Dynamic crossover in the persistence probability of manifolds at criticality. J. Stat. Mech. P12029 (2010)

  17. Hammersley, J.M.: Harnesses. In: Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability, vol. III: Physical Sciences, pp. 89–117. University of California Press, Berkeley, California (1965/1966)

  18. Krug, J., Kallabis, H., Majumdar, S.N., Cornell, S.J., Bray, A.J., Sire, C.: Persistence exponents for fluctuating interfaces. Phys. Rev. E. 56(3), 2702–2712 (1997)

    Article  Google Scholar 

  19. Lamperti, J.: Semi-stable stochastic processes. Trans. Am. Math. Soc. 104(1), 62–78 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, W.V., Shao, Q.M.: Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1, 95–106 (2005)

    Article  MathSciNet  Google Scholar 

  21. Molchan, G.: Survival exponents for some Gaussian processes. Int. J. Stoch. Anal. 137271, 1–20 (2012)

    MATH  MathSciNet  Google Scholar 

  22. Majumdar, S.N., Bray, A.J.: Spatial persistence of fluctuating interfaces. Phys. Rev. Lett. 86(17), 3700–3703 (2001)

    Article  Google Scholar 

  23. Majumdar, S.N., Bray, A.J.: Persistence of manifolds in non-equilibrium critical dynamics. Phys. Rev. Lett. 91(3), 030602 (2003)

    Article  Google Scholar 

  24. Pickands, J.: Asymptotic properties of maximum in a stationary Gaussian processes. Trans. Am. Math. Soc. 145, 75–86 (1969)

    MATH  MathSciNet  Google Scholar 

  25. Newell, G.F., Rosenblatt, M.: Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Stat. 33(4), 1306–1313 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sakagawa, H.: Persistence probability for a class of Gaussian processes related to random interface models. Adv. Appl. Probab. 47(1), 146–163 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shur, M.G.: On the maximum of a Gaussian stationary process. Theor. Probab. Appl. 10, 354–357 (1965)

    Article  Google Scholar 

  28. Slepian, D.: The one-sided barrier problem for Gaussian noise. Bell Syst. Tech. 41, 463–501 (1962)

    Article  MathSciNet  Google Scholar 

  29. Schehr, G., Majumdar, S.N.: Real roots of random polynomials and zero crossing properties of diffusion equation. J. Stat. Phys. 132(2), 235–273 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Uhlenbeck, G.E., Ornstein, L.S.: On the theory of Brownian motion. Phys. Rev. Lett. 36, 823–841 (1930)

    MATH  Google Scholar 

  31. Unterberger, J.: Stochastic calculus for fractional Brownian motion with Hurst exponent \(H>\frac{1}{4}\): A rough path method by analytic extension. Ann. Probab. 37(2), 565–614 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lawler, G.F., Limic, V.: Random walk: a modern introduction. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  33. Watanabe, H.: An asymptotic property of Gaussian stationary processes. Proc. Jpn. Acad. 44, 895–896 (1968)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research is the outgrowth of discussions with H. Sakagawa during a research visit of A. D. that was funded by T. Funaki from Tokyo University. We are indebted to H. Sakagawa for sharing with us a preprint of [26], to J. Ding for an alternative proof of Theorem 1.2(b) and to O. Zeitouni for helpful discussions. We thank G. Schehr for bringing the references [5, 16] to our notice and the referees whose suggestions much improved this article.

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Correspondence to Sumit Mukherjee.

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A. Dembo: Research partially supported by NSF Grant DMS-1106627.

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Dembo, A., Mukherjee, S. Persistence of Gaussian processes: non-summable correlations. Probab. Theory Relat. Fields 169, 1007–1039 (2017). https://doi.org/10.1007/s00440-016-0746-9

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  • DOI: https://doi.org/10.1007/s00440-016-0746-9

Keywords

  • Persistence probabilities
  • Gaussian processes
  • Regularly varying
  • \(\nabla \phi \)-interface

Mathematics Subject Classification

  • Primary 60G15
  • Secondary 82C24