15. A Potential Theoretic Approach to Tanaka Formula for Asymmetric Lévy Processes

In this paper, we shall introduce the Tanaka formula from viewpoint of the Doob–Meyer decomposition. For symmetric Lévy processes, if the local time exists, Salminen and Yor (Tanaka formula for symmetric Lévy processes. In: Séminaire de Probabilités XL. Lecture notes in mathematics, vol. 1899, Springer, Berlin, pp. 265–285, 2007) obtained the Tanaka formula by using the potential theoretic techniques. On the other hand, for strictly stable processes with index α ∈ (1, 2), we studied the Tanaka formula by using Itô’s stochastic calculus and the Fourier analysis. In this paper, we study the Tanaka formula for asymmetric Lévy 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 Lévy processes, and it generalized the result in Yano (J Math Ind 5(A):17–24, 2013).