We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short- and long-range dependent components. Applying this theorem, we obtain a homogenization result for a slow/fast system driven by such Hermite noises.

中文翻译：

---

具有 Hermite 过程移动平均快速衰减的幂级数的泛函极限定理

我们的目标是获得与分数（可能是非高斯）噪声相互作用的无源示踪剂的均匀化定理。为此，我们分析了 Hermite-Volterra 过程的归一化泛函的极限定理，并将现有结果扩展到涵盖具有快速衰减系数的幂级数。在短程依赖的情况下，我们要么收敛到 Wiener 过程，要么在长程依赖的情况下收敛到 Hermite 过程。此外，我们证明了在具有短期和长期依赖分量的多变量情况下的收敛性。应用这个定理，我们获得了由这种 Hermite 噪声驱动的慢/快系统的均匀化结果。