Principal values of Brownian local times and their related topics

The study of the Cauchy’s principal value of Brownian local times has started with Ito,K. and McKean,H.P.Jr. [19]. In this article, we would like to see how their idea combined with a generalization of the Ito formula in stochastic calculus has grown up to be an interesting branch in the study of Brownian motion. After Ito-McKean, Ezawa,H.,Klauder,J.R. and Shepp,A. ([10]) discussed principal values of local times in an investigation of vestigical effects of the singular potential λV(x) = λ|x−c|− α where λ → 0 in the Feynmann-Kac integral. It seems remarkable that in the early 70’s, when few payed attention to these notions, they have been already used for some physical purposes by physicists cooperating with a mathematician. From the end of the 70’s to the beginning of the 80’s two studies have been published, one by Fukushima,M. [14] and the other by Föllmer,H. [13], and they have played the role of the corner stone in the development of a new branch in stochastic analysis. Föllmer has proposed stochastic analysis of Dirichlet process which is a generalization of semi-martingale. The class of additive functionals of zero energy has been investigated by Fukushima in the scheme of symmetric Hunt processes. These additive functionals can not be treated in the frame of the theory of semi-martingales. The studies have been essentially concerned with generalizations of the Ito-formula. Owing to their propositions, we have rich possibilities to apply the Ito calculus to wider class of processes than that of semi-martingales.