Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond

The goal of this paper is threefold. First, we survey the existing results on Hunt’s hypothesis (H) for Markov processes and Getoor’s conjecture for L'{e}vy processes. Second, we investigate (H) for multidimensional L'{e}vy processes from the viewpoints of projections and energy, respectively. Third, we present a few open questions for further study.