# Moderate Deviation for Random Elliptic PDEs with Small Noise

## TL;DR

This paper analyzes the probability of rare events in elliptic PDEs with small random noise, providing sharp asymptotic approximations for systems with low but non-negligible uncertainty.

## Contribution

It introduces a novel asymptotic analysis framework for rare events in elliptic PDEs with diminishing noise levels, advancing understanding of low-noise stochastic systems.

## Key findings

- Derived sharp asymptotic approximations for rare event probabilities
- Characterized the impact of small noise on PDE solution uncertainty
- Provided theoretical insights into low-noise elliptic PDE behavior

## Abstract

Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from, e.g., 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 exists. We develop sharp approximations of the probability of a large class of rare events.