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Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium) Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Hironobu Sakagawa
Hironobu Sakagawa*
Affiliation:
Keio University
*
∗ Postal address: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama, 223-8522, Japan. Email address: [email protected]
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Abstract

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We consider a class of Gaussian processes which are obtained as height processes of some (d + 1)-dimensional dynamic random interface model on ℤd. We give an estimate of persistence probability, namely, large T asymptotics of the probability that the process does not exceed a fixed level up to time T. The interaction of the model affects the persistence probability and its asymptotics changes depending on the dimension d.

Keywords

Persistence probabilityrandom interfaceGaussian processinteracting diffusion process

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C41: Dynamics of random walks, random surfaces, lattice animals, etc.
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 146 - 163
Copyright
© Applied Probability Trust 

References

Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Aurzada, F. (2011). On the one-sided exit problem for fractional Brownian motion. Electron. Commun. Prob. 16, 392–404.Google Scholar
Aurzada, F. and Baumgarten, C. (2011). Survival probabilities of weighted random walks. ALEA Latin Amer. J. Prob. Math. Statist. 8, 235–258.Google Scholar
Aurzada, F. and Dereich, S. (2013). Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. H. Poincaré Prob. Statist. 49, 236–251.CrossRefGoogle Scholar
Aurzada, F. and Simon, T. (2012). Persistence probabilities & exponents. Preprint. Available at http://uk.arxiv.org/abs/1203.6554.Google Scholar
Bolthausen, E., Deuschel, J.-D. and Giacomin, G. (2001). Entropic repulsion and the maximum of two dimensional harmonic crystal. Ann. Prob. 29, 1670–1692.Google Scholar
Bramson, M. and Zeitouni, O. (2012) Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Commun. Pure Appl. Math. 65, 1–20.CrossRefGoogle Scholar
Dembo, A. and Deuschel, J.-D. (2007). Aging for interacting diffusion processes. Ann. Inst. H. Poincaré Prob. Statist. 43, 461–480.CrossRefGoogle Scholar
Dembo, A. and Mukherjee, S. (2015). No zero-crossings for random polynomials and the heat equation. Ann. Prob. 43, 85–118.CrossRefGoogle Scholar
Dembo, A., Ding, J. and Gao, F. (2013). Persistence of iterated partial sums. Ann. Inst. H. Poincaré Prob. Statist. 49, 873–884.Google Scholar
Deuschel, J.-D. (1989). Invariance principle and empirical mean large deviations of the critical Ornstein–Uhlenbeck process. Ann. Prob. 17, 74–90.Google Scholar
Deuschel, J.-D. (2005). The random walk representation for interacting diffusion processes. In Interacting Stochastic Systems. Springer, Berlin, pp. 377–393.CrossRefGoogle Scholar
Ferrari, P. A., Fontes, L. R. G., Niederhauser, B. M. and Vachkovskaia, M. (2004). The serial harness interacting with a wall. Stoch. Process. Appl. 114, 175–190.Google Scholar
Funaki, T. (2005). Stochastic interface models. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1869). Springer, Berlin, pp. 103–274.Google Scholar
Garet, O. (2000). Infinite dimensional dynamics associated to quadratic Hamiltonians. Markov Process. Relat. Fields 6, 205–237.Google Scholar
Hammersley, J. M. (1967). Harnesses. In Proc. Fifth Berkeley Symp. Math. Statist. Prob. Vol. III. Physical Sciences. University of California Press, Berkeley, CA, pp. 89–117.Google Scholar
Krug, J. et al. (1997). Persistence exponents for fluctuating interfaces. Phys. Rev. E 56, 2702–2712.Google Scholar
Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge University Press.Google Scholar
Li, W. V. and Shao, Q.-M. (2004). Lower tail probabilities for Gaussian processes. Ann. Prob. 32, 216–242.Google Scholar
Majumdar, S. N. (1999). Persistence in nonequilibrium systems. Current Sci. 77, 370–375.Google Scholar
Molchan, G. M. (1999). Maximum of a fractional Brownian motion: probabilities of small values. Commun. Math. Phys. 205, 97–111.Google Scholar
Newell, G. F. and Rosenblatt, M. (1962). Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Statist. 33, 1306–1313.Google Scholar
Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.Google Scholar
Takeuchi, K. A. and Sano, M. (2012). Evidence for geometry-dependent universal fluctuations of the Kardar–Parisi–Zhang interfaces in liquid-crystal turbulence. J. Statist. Phys. 147, 853–890.Google Scholar
Uchiyama, K. (1980). Brownian first exit from and sojourn over one-sided moving boundary and application. Z. Wahrscheinlichkeitsth. 54, 75–116.Google Scholar