An integration of satellites and terrestrial networks is crucial for enhancing performance of next-generation communication systems. However, the networks are hindered by the long-distance path loss and security risks in urban canyons. In this work, we propose a satellite-terrestrial covert communication system assisted by the aerial active transmissive reconfigurable intelligent surface (AAT-RIS) to improve the channel capacity while ensuring the transmission covertness. Specifically, we first derive the minimal detection error probability (DEP) under the worst condition that the Warden has perfect channel state information. Then, we formulate an AAT-RIS-assisted satellite-terrestrial covert communication optimization problem (ASCCOP) to maximize the sum of the fair channel capacity for all ground users while meeting the strict covert constraint, by jointly optimizing the trajectory and active beamforming of the AAT-RIS. Due to the challenges posed by the complex and high-dimensional state-action spaces as well as the need for efficient exploration in dynamic environments, we propose a generative deterministic policy gradient (GDPG) algorithm, which is a generative deep reinforcement learning-based method to solve the online ASCCOP. Concretely, the generative diffusion model is utilized as the policy representation of the proposed algorithm to enhance the exploration process by generating diverse and high-quality samples through a series of denoising steps. Moreover, we incorporate an action gradient mechanism to accomplish the policy improvement of the proposed algorithm, which refines the better state-action pairs through the gradient ascent. Simulation results demonstrate that the proposed approach significantly outperforms important benchmarks, and also validate the robustness under different algorithm parameters and environment settings.

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