AI 中文总结
本文在齐次李群上建立加权非齐次Besov空间的统一理论，将其应用于含Rockland算子与奇异初始数据的抛物Anderson型方程，分别在时空噪声和纯空间噪声下得到相关结果，依赖温和缝合引理完成推导。
AI 中文摘要
我们在一般齐次李群上发展了内在的加权非齐次Besov空间理论，无需借助特定群的论证。从基于局部测试函数的定义出发，我们建立了等价的类小波多尺度刻画，这为推导Besov嵌入、泰勒余项刻画、类Young乘积估计、卷积半群的Schauder估计以及随机分布的加权Kolmogorov准则提供了统一机制。我们将该框架应用于与正Rockland算子及奇异初始数据相关的抛物Anderson型方程：对于时空噪声，我们的结果覆盖Young regime；对于纯空间噪声，我们还利用Rockland算子的Cole-Hopf变换变体，在第一奇异 regime 中建立了适定性。两项应用均依赖于适配随时间变化的Banach空间及初始时刻含奇异性增量的温和缝合引理。
英文摘要
We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions. We apply this framework to parabolic Anderson-type equations associated with positive Rockland operators and with singular initial data. For space-time noise, our results cover the Young regime. For purely spatial noise, we also establish well-posedness in the first singular regime using a variant of the Cole-Hopf transform for Rockland operators. Both applications rely on a mild sewing lemma that accommodates time-dependent Banach spaces and increments with a singularity at the initial time.
