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Extreme Analysis of a Random Ordinary Differential Equation

Part of: Limit theorems

Published online by Cambridge University Press:  30 January 2018

Jingchen Liu  and
Xiang Zhou
Jingchen Liu*
Affiliation:
Columbia University
Xiang Zhou*
Affiliation:
City University of Hong Kong
*
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA. Email address: jcliu@stat.columbia.edu
∗∗ Postal address: Y6524 (Yellow Zone), 6/F Academic 1, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong.
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Abstract

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In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

Keywords

Random differential equationextremesrare event

MSC classification

Primary: 60F10: Large deviations
Secondary: 65Z05: Applications to physics
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1021 - 1036
Copyright
© Applied Probability Trust 

References

Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.Google Scholar
Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Adler, R. J., Blanchet, J. and Liu, J. (2008). Efficient simulation for tail probabilities of Gaussian random fields. In Proc. Winter Simulation Conf. 2008, IEEE, New York, pp. 328–336.Google Scholar
Adler, R. J., Blanchet, J. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Prob. 22, 1167–1214.Google Scholar
Azaı¨s, J.-M. and Wschebor, M. (2008). A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stoch. Process. Appl. 118, 1190–1218. (Erratum: 120 (2010), 2100–2101.)Google Scholar
Borell, C. (2003). The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337, 663–666.Google Scholar
Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Prob. 40, 1069–1104.Google Scholar
Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40, 262–293.Google Scholar
Liu, J. and Xu, G. (2014). On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields. Ann. Appl. Prob. 24, 1691–1738.Google Scholar
Liu, J., Zhou, X., Patra, R. and Weinan, E. (2011). Failure of random materials: a large deviation and computational study. In Proc. Winter Simulation Conf. 2011, IEEE, New York, pp. 3779–3789.Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI.Google Scholar
Taylor, J. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Prob. 31, 533–563.Google Scholar
Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Prob. 33, 1362–1396.Google Scholar
Cirel'son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan–USSR Symposium on Probability Theory (Lecture Notes Math. 550), Springer, Berlin, pp. 20–41.Google Scholar
URL: http://stat.columbia.edu/\verb+ +jcliu/paper/OneDimDirichletDensity26FinalSupplement.pdf.Google Scholar
URL: http://arxiv.org/abs/1309.3851.Google Scholar