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Persistence of Gaussian processes: non-summable correlations

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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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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Building 380, Sloan Hall, Stanford, CA, 94305, USA

    Amir Dembo

  2. Department of Statistics, Columbia University, 1011 SSW, 1255 Amsterdam Avenue, New York, NY, 10027, USA

    Sumit Mukherjee

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  1. Amir Dembo
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  2. Sumit Mukherjee
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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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