Moderate deviation for random elliptic PDE with small noise

Xiaoou Li; Jingchen Liu; Jianfeng Lu; Xiang Zhou

doi:10.1214/17-AAP1373

Moderate deviation for random elliptic PDE with small noise

Xiaoou Li, Jingchen Liu, Jianfeng Lu, Xiang Zhou

Statistics (Twin Cities)

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Abstract

Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from imprecise measurement and intrinsic randomness. In this paper, we perform asymptotic rare-event analysis for such elliptic PDEs with random inputs. In particular, we consider the asymptotic regime that the noise level converges to zero suggesting that the system uncertainty is low, but does exist. We develop sharp approximations of the probability of a large class of rare events.

Original language	English (US)
Pages (from-to)	2781-2813
Number of pages	33
Journal	Annals of Applied Probability
Volume	28
Issue number	5
DOIs	https://doi.org/10.1214/17-AAP1373
State	Published - Oct 2018

Bibliographical note

Funding Information:
Received October 2016; revised November 2017. 1Supported in part by the National Science Foundation (DMS-1712657). 2Supported in part by the National Science Foundation (SES-1323977, IIS-1633360) and Army Grant (W911NF-15-1-0159). 3Supported in part by National Science Foundation (DMS-1454939). 4Supported by Hong Kong General Research Fund (109113, 11304314, 11304715). MSC2010 subject classifications. Primary 60F10, 60Z05; secondary 60G15. Key words and phrases. Random partial differential equation, rare event, moderate deviation.

Publisher Copyright:
© Institute of Mathematical Statistics, 2018.

Keywords

Moderate deviation
Random partial differential equation
Rare event

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abstract = "Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from imprecise measurement and intrinsic randomness. In this paper, we perform asymptotic rare-event analysis for such elliptic PDEs with random inputs. In particular, we consider the asymptotic regime that the noise level converges to zero suggesting that the system uncertainty is low, but does exist. We develop sharp approximations of the probability of a large class of rare events.",

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Moderate deviation for random elliptic PDE with small noise

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