Sequential reactivation (replay) of neurons in the hippocampus has been widely observed in sharp wave ripple events (SWRs) during awake and sleep periods, and they are associated with mnemonic processes, such as memory retrieval, consolidation, future planning and decision making. Previous studies have shown that the replay trajectories in the rat hippocampus are rather random, which resemble at least two different types of stochastic process, one corresponding to Brownian diffusion discovered during sleep SWRs and the other Lévy superdiffusion discovered during awake SWRs. The underlying mechanism of generating these different types of replay trajectory remains, however, obscure. In this study, we build a continuous attractor neural network (CANN) for encoding spatial information in the hippocampal circuit, and show that neural adaptation, exemplified by spike frequency adaptation (SFA) here, can serve as a general mechanism to implement the two types of replay trajectories. In our model, the role of SFA is to induce a negative feedback to neuronal responses, which destabilizes the bump-like network state. Specifically, when the SFA strength is sufficiently large, it induces a travelling wave state of the bump, i.e., the bump moves spontaneously in the attractor space. Thus, by modulating the SFA strength, we can observe two different stochastic processes of the bump movement. When the SFA strength is small, the bump movement is mainly driven by noise fluctuations, and its trajectory displays Brownian motion; while when the SFA strength is large, the bump movement is driven by both noise fluctuations and the intrinsic mobility (travelling wave) induced by SFA, and its trajectory displays Lévy superdiffusion. We carry out theoretical analyses and simulations to demonstrate that our model reproduce various experimental findings, including the Brownian motion in sleep SWRs, the Lévy motion in awake SWRs, and the anti-phase locking between neural activities and long-jump motions in the awake replay trajectories. We hope that this study helps us to understand the neural mechanism for generating rich dynamics in the hippocampus.