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Persistence Exponents for Gaussian Random Fields of Fractional Brownian Motion Type

  • G. Molchan
Article
  • 27 Downloads

Abstract

The fractional Brownian motion of index 0 < H < 1, H-FBM, with d-dimensional time is considered on an expanding set \( T\Delta \), where \( \Delta \) is a bounded convex domain that contains 0 at its boundary. The main result: if 0 is a point of smoothness of the boundary, then the log-asymptotics of the probability that H-FBM does not exceed a fixed positive level in \( T\Delta \) is \( (H - d + o(1))\log T,T \to \infty \). Some generalizations of this result to isotropic but not self-similar Gaussian fields with stationary increments are also considered.

Keywords

Fractional Brownian motion Persistence probability One-sided exit problem 

Notes

Acknowledgements

I am very grateful to two anonymous reviewers for a careful reading of the manuscript and for their suggestions for improving the presentation. This research was supported by the Russian Science Foundation through the Research Project 17-11-01052.

References

  1. 1.
    Aurzada, F., Dereich, S.: Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincar´e Probab. Stat. 49(1), 236–251 (2013)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Aurzada, F., Simon, T.: Persistence Probabilities and Exponents. Lecture Notes in Mathematics, vol. 2149, pp. 183–221. Springer, New York (2015)zbMATHGoogle Scholar
  3. 3.
    Aurzada, F., Monch, C.: Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes. Theory Probab. Appl., to appears. Preprint available in https: arXiv:1607.045045 (2016)
  4. 4.
    Aurzada, F., Guillotin-Plantard, N., Pene, F.: Persistence probabilities for stationary increment processe. Stoch. Process. Appl. 128, 1750–1771 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aurzada, F., Buck, M.: Persistence probabilities of two-sided (integrated) sums of correlated stationary Gaussian sequences. J. Stat. Phys. 170, 784–799 (2018)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bertoin, J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193, 397–406 (1998)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bray, A.J., Majumdar, S.N., Schehr, G.: Persistence and first-passage properties in non-equilibrium systems. Adv. Phys. 62(3), 225–361 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Dembo, A., Ding, J., Gao, F.: Persistence of iterated partial sums. Ann. Inst. Henri Poincare Probab. Stat. 49(3), 873–884 (2013)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fernique, X.: Regularite des Trajectories des Functions Aleatoires Gaussiannes. Lecture Notes in Mathematics, vol. 480. Springer, Berlin (1975)zbMATHGoogle Scholar
  10. 10.
    Inozemcev, O., Marchenko, V.: On majorants of genus zero. Uspehi Mat. Nauk. 11, 173–178 (1956)MathSciNetGoogle Scholar
  11. 11.
    Lifshits, M.: Lectures on Gaussian Processes. Springer, New York (2012)CrossRefGoogle Scholar
  12. 12.
    Molchan, G.: Maximum of fractional Brownian motion: probabilities of small values. Commun. Math. Phys. 205(1), 97–111 (1999)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Molchan, G.: Unilateral small deviations of processes related to the fractional Brownian motion. International Conference: Small deviations probabilities and related topic. St Petersburg (2005)Google Scholar
  14. 14.
    Molchan, G.: Survival exponents for fractional Brownian motion with multivariate time. ALEA, Lat. Am. J Probab. Math. Stat. 14, 1–7 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Molchan, G.: The inviscid Burgers equation with fractional Brownian initial data: the dimension of regular Lagrangian points. J. Stat. Phys. 167(6), 1546–1554 (2017)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Profeta, C., Simon, T.: Persistence of integrated stable processes. Probab. Theory Relat. Fields 162(3–4), 463–485 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sinai, Y.G.: Statistics of shocks in solutions of the inviscid Burgers equation. Commun. Math. Phys. 148, 601–621 (1992)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    She, Z., Aurell, E., Frish, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–642 (1992)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Samorodnitsky, G., Taqqu, M.: Stable non-Gaussian Random Processes. Chapman and Hall, New York (1994)zbMATHGoogle Scholar
  20. 20.
    Whitt, W.: Stochastic-Process Limits. Operations Research and Financial Engineering. Springer, New York (2002)zbMATHGoogle Scholar
  21. 21.
    Yaglom, A.: Correlation Theory of Stationary and Related Random Functions, vol. 12. Springer, New York (1987)zbMATHGoogle Scholar
  22. 22.
    Yosida, K.: Functional Analysis. Academic Press, New York (1968)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Earthquake Prediction Theory and Mathematical GeophysicsRussian Academy of ScienceMoscowRussian Federation
  2. 2.National Research University Higher School of EconomicsMoscowRussian Federation

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