Multi-level implicit discourse relation recognition (MIDRR) is a challenging task to recognize the hierarchical discourse relations between the arguments with the absence of connectives. Recent methods tend to incorporate the static hierarchical structure containing all senses (defined as global hierarchy) into prompt tuning through a path prompt template or hierarchical label refining. Howerver, hierarchical modeling is independent of the verbalizer, resulting in a failure to effectively utilize the output probability distribution information of verbalizer. Besides, they ignore the utilization of the dynamic hierarchical label sequence for each instance (defined as local hierarchy) in prompt tuning. In this paper, we propose a global and local hierarchical prompt tuning (GLHPT) framework, which utilize prior knowledge of PLMs while better incorporating hierarchical information from two aspects. We leverage bottom-up propagated probability as the global hierarchy to inject it into multi-level verbalizer (MLV). Furthermore, we design a local hierarchy-driven contrastive learning (LHCL) to improve the probability distribution of MLV. Finally, our model achieves competitive results on two benchmacks.