具有非线性保守噪声的某些约束波方程的小质量极限

Sandra Cerrai, Mengzi Xie
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引用次数: 0

摘要

我们研究了一个随机阻尼波方程系统的小质量极限,也称为斯莫卢霍夫斯基-克拉默扩散近似(见 \cite{kra} 和 \cite{smolu}),其解受限于区间$(0,L)$上平方可积分函数空间的单元球内。随机扰动由非线性乘法高斯噪声给出,其中的随机微分从斯特拉顿维奇意义上理解。由于其特殊的结构,这种噪声不仅保留了$\mathbb{P}$-a.s. 约束,而且还保留了一个合适的能量函数。在极限中,我们推导出一个确定性系统,它仍然限定在 $L^2$ 的单位球内,但包含附加项。这些项取决于噪声的再现核,并考虑了约束与我们选择的特定保守噪声之间的相互作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The small-mass limit for some constrained wave equations with nonlinear conservative noise
We study the small-mass limit, also known as the Smoluchowski-Kramers diffusion approximation (see \cite{kra} and \cite{smolu}), for a system of stochastic damped wave equations, whose solution is constrained to live in the unitary sphere of the space of square-integrable functions on the interval $(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative Gaussian noise, where the stochastic differential is understood in Stratonovich sense. Due to its particular structure, such noise not only conserves $\mathbb{P}$-a.s. the constraint, but also preserves a suitable energy functional. In the limit, we derive a deterministic system, that remains confined to the unit sphere of $L^2$, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the particular conservative noise we choose.
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