By constructing a coupling with unbounded time-dependent drift, a lower bound estimate of dimension-free Harnack inequality with power is obtained for a large class of stochastic differential equation with multiplicative noise. The key is an application of the inverse Hölder inequality. Combining this with the well-known upper bound, bilateral dimension-free Harnack inequality with power is established. As a dual inequality, the bilateral shift-Harnack inequalities with power are also investigated for stochastic differential equation with additive noise. Applications to the study of heat kernel inequalities are provided to illustrate the obtained inequalities.

Reliable energy forecasting is essential for the planning and dispatch of power and fuel systems; however, energy series are often short and exhibit pronounced nonlinearity. To tackle this small sample setting, we propose a gray random vector functional link (GRVFL) framework and further derive a kernelized variant (KGRVFL). In GRVFL, an RVFL network is integrated into gray system modeling, and the parameters are learned via sparsity-regularized regression, enabling stable and reproducible training without backpropagation or evolutionary optimization. Hyperparameters are tuned using Bayesian optimization driven by a Top-k mean absolute percentage error (Top-k MAPE) criterion to improve robustness. To further promote compactness, we introduce a fractional ratio-type Fr-$\ell 1$ penalty and solve the resulting problem efficiently using a fractional coordinate descent (FCD) algorithm. The proposed methods are assessed on six real-world energy datasets using eight evaluation metrics. Comparisons with nine gray model baselines and six machine learning forecasters demonstrate that the sparse KGRVFL (SKGRVFL) achieves higher predictive accuracy and improved training stability under small sample conditions.

We consider the Cox–Ingersoll–Ross (CIR) model in time-dependent domains, that is, the CIR process in time-dependent domains reflected at the time-dependent boundary. This is a very meaningful question, as the CIR model is commonly used to describe interest rate models, and interest rates are often artificially set within a time-dependent domain by policy makers. We consider the most fundamental question of recurrence versus transience for normally reflected CIR process with time-dependent domains, and we examine some precise conditions for recurrence versus transience in terms of the growth rates of the boundary. The drift terms and the diffusion terms of the CIR processes in time-dependent domains are carefully provided. In the transience case, we also investigate the last passage time, while in the case of recurrence, we also consider the positive recurrence of the CIR processes in time-dependent domains.