Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

On the failure probability of one dimensional random material under delta external force

Pages: 499 – 521



Jingchen Liu (Department of Statistics, Columbia University, New York, N.Y., U.S.A.)

Xiang Zhou (Division of Applied Mathematics, Brown University, Providence, R.I., U.S.A.)


We provide an asymptotic analysis of the small failure probabilities for a piece of elastic random material under a certain external force and boundary condition. The displacement of the material is described by a one dimensional stochastic elliptic differential equation. The differential equation admits random coefficients described by a Gaussian process. Failure is defined as the event that the maximum strain of the material exceeds a certain level. We derive asymptotic approximations of the probability that the strain exceeds a high level $b$ that tends to infinity.


Gaussian random field, stochastic elliptic partial differential equation, maximum strain, failure probability

2010 Mathematics Subject Classification

60G15, 60H15

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