带输运噪声的随机偏微分方程的大偏差原理（LDP）与中心极限定理（CLT）
LDP and CLT for SPDEs with Transport Noise
AI总结：
本文研究带输运噪声的随机偏微分方程的标度极限行为，针对高维随机线性输运方程和二维涡量形式Euler方程，建立了带强收敛性与显式速率的中心极限定理，运用超临界输运方程可解性、$\Gamma$-收敛等工具分析其大偏差与高斯涨落特性。
摘要：
本文研究带输运噪声的随机偏微分方程的解，已知这类解在合适的标度极限下会收敛到带附加粘性项的对应确定性偏微分方程的解。我们针对两个感兴趣的情形探究了该标度极限背后的大偏差和高斯涨落：维度$D\geq 2$的随机线性输运方程，以及涡量形式的二维Euler方程。在两种情形下，我们都建立了具有强收敛性和显式收敛速率的中心极限定理。证明过程依赖于非平凡的工具，例如带超临界系数的输运方程的可解性，以及$\Gamma$-收敛论证。
英文摘要：
In this work we consider solutions to stochastic partial differential equations with transport noise, which are known to converge, in a suitable scaling limit, to solution of the corresponding deterministic PDE with an additional viscosity term. Large deviations and Gaussian fluctuations underlying such scaling limit are investigated in two cases of interest: stochastic linear transport equations in dimension $D\geq 2$ and $2$D Euler equations in vorticity form. In both cases, a central limit theorem with strong convergence and explicit rate is established. The proofs rely on nontrivial tools, like the solvability of transport equations with supercritical coefficients and $Γ$-convergence arguments.
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