Local large deviations for randomly forced nonlinear wave equations with localized damping

Yuxuan Chen, Ziyu Liu, Shengquan Xiang, Zhifei Zhang
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Abstract

We study the large deviation principle (LDP) for locally damped nonlinear wave equations perturbed by a bounded noise. When the noise is sufficiently non-degenerate, we establish the LDP for empirical distributions with lower bound of a local type. The primary challenge is the lack of compactness due to the absence of smoothing effect. This is overcome by exploiting the asymptotic compactness for the dynamics of waves, introducing the concept of asymptotic exponential tightness for random measures, and establishing a new LDP approach for random dynamical systems.
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具有局部阻尼的随机强迫非线性波方程的局部大偏差
我们研究了受有界噪声扰动的局部阻尼非线性波方程的大偏差原理(LDP)。当噪声足够非退化时,我们建立了具有局部类型下限的经验分布的大偏差原理。主要的挑战是由于没有平滑效应而缺乏紧凑性。我们利用波动力学的渐近紧凑性,引入随机度量的渐近指数紧密性概念,建立了随机动力系统的新 LDP 方法,从而克服了这一难题。
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