On discontinuous additive functionals and Lévy measures of a Markov process

Abstract

Introduction.1) In M. Motoo-S. Watanabe [5] we have investigated the class m of square-integrable additive functionals which are martingales. The first purpose of this paper is to determine the structure of the subspace md which is defined as the orthogonal complement of the space mc formed of all continuous elements of m. As a corollary we can obtain the general form of the quasi-left continuous purely discontinuous additive functionals which was first obtained by M. Motoo.2) At the same time we see that we can give a definition of the Levy measure by means of these additive functionals and we shall discuss some properties of it. The author wishes to express his hearty thanks to Prof. M. Motoo for his kind encouragement and valuable suggestions.