A potential theoretic approach to Tanaka formula for asymmetric L\'evy processes

Abstract

In this paper, we shall introduce the Tanaka formula from viewpoint of the Doob-Meyer decomposition. For symmetric L\'evy processes, if the local time exists, Salminen and Yor (2007) obtained the Tanaka formula by using the potential theoretic techniques. On the other hand, for asymmetric stable processes with index $\alpha \in (1,2)$, we studied the Tanaka formula by using It\^o's stochastic calculus and the Fourier analysis. In this paper, we study the Tanaka formula for asymmetric L\'evy processes via the potential theoretic approach. We give several examples for important processes. Our approach also gives the invariant excessive function with respect to the killed process in the case of asymmetric L\'evy processes and it generalized the result in Yano (2013).