Nekhoroshev stability for random generalized Hamiltonian systems with different regularities

In this article, we establish the Nekhoroshev stability of nearly integrable generalized Hamiltonian systems with bounded random perturbations possessing different regularity conditions. We generalize the original framework for proving the Nekhoroshev theorem. Using this unified framework, we can derive different normal form lemmas based on various regularity conditions, leading to results for stability times of different scales. Furthermore, this method allows perturbation functions with a certain degree of randomness and can be applied within the context of generalized Hamiltonian systems.