Generalized refracted L\'evy process and its application to exit problem

Generalizing Kyprianou--Loeffen's refracted L\'evy processes, we define a new refracted L\'evy process which is a Markov process whose positive and negative motions are L\'evy processes different from each other. To construct it we utilize the excursion theory. We study its exit problem and the potential measures of the killed processes. We also discuss approximation problem.