AI 中文总结
该研究针对环面上带快速振荡系数的McKean--Vlasov随机反应扩散方程建立三个平均原理，证明解的均方收敛、有界整体解的均方距离一致消失及拉回吸引子上半连续收敛到全局吸引子，并给出相关应用模型。
AI 中文摘要
我们针对维数 $d\le3$ 的环面 $\mathbb T^d$ 上具有快速振荡系数的分布依赖随机反应扩散方程，建立了三个平均原理。首先，方程的解在有限时间区间上依均方一致收敛于平均方程的解。在压缩条件下，原方程与平均方程都存在唯一的有界整体解，二者的均方距离对所有 $t\in\mathbb R$ 一致趋于零。在概率律层面，原非自治方程具有一族拉回吸引子，而平均方程拥有一个全局吸引子；前者在系数包上一致地依上半连续收敛于后者。作为应用，我们提出了一类由大规模相互作用系统启发的随机反应扩散模型。
英文摘要
We establish three averaging principles for distribution-dependent stochastic reaction--diffusion equations with rapidly oscillating coefficients on the torus $\mathbb T^d$, $d\le3$. First, solutions converge in mean square, uniformly on finite time intervals, to solutions of the averaged equation. Under a contraction condition, both the original and averaged equations admit unique bounded entire solutions whose mean-square distance vanishes uniformly for all $t\in\mathbb R$. At the level of probability laws, the original nonautonomous equation possesses a family of pullback attractors, whereas the averaged equation has a global attractor; the former converge upper-semicontinuously to the latter, uniformly over the coefficient hull. As an application, we present a class of stochastic reaction--diffusion models motivated by large-scale interacting systems.
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