Brownian movement and the Tanaka formula in analysis

In our previous paper [5], we have obtained a decomposition of \f\, where f is a function defined on R-d, that is analogous to the one proved by H. Tanaka in the early sixties for Brownian martingales (the so-called 'Tanaka formula'). The original proofs use purely analytic methods (e.g. the Calderon-Zygmund theory, etc.). In this paper, we give a new proof of our 'Tanaka formula in analysis', that is based on probabilistic arguments. The main tools here are Brownian motion, stochastic calculus and Burkholder-Gundy inequalities for martingales. These methods allow us to improve somewhat our previous results, by proving that some significant constants do not depend on the dimension d.