带非线性保守噪声的若干约束波动方程的小质量极限
The small-mass limit for some constrained wave equations with nonlinear conservative noise
AI总结：
本文研究带Stratonovich型非线性乘性保守高斯噪声的约束随机阻尼波动方程的小质量极限，推导出仍约束于L²单位球面、含噪声再生核相关附加项的确定性系统，揭示约束与保守噪声的相互作用。
摘要：
我们研究一类随机阻尼波动方程组的小质量极限，也称为Smoluchowski-Kramers扩散近似（见文献\cite{kra}和\cite{smolu}），其解被约束在区间$(0,L)$上平方可积函数空间的单位球面内。随机扰动由非线性乘性高斯噪声给出，其中随机微分按Stratonovich意义理解。由于其特殊结构，这类噪声不仅在$\u2119$-几乎必然意义下保持约束，还能保持一个合适的能量泛函。在极限情形下，我们推导出一个确定性系统，该系统仍被限制在$L^2$的单位球面内，但包含附加项。这些项依赖于噪声的再生核，用于解释约束与我们选取的特定保守噪声之间的相互作用。
英文摘要：
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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