On some representations of continuous additive functionals locally of zero energy (Q795409)

If \(X_ t\) is a diffusion process and u is a \(C^ 2\)-function, then by Ito formula, \(u(X_ t)-u(X_ 0)\) can be uniquely decomposed as ''a local martingale \(+\) a process of locally bounded variation''. \textit{M. Fukushima} [Nagoya Math. J. 74, 137-168 (1979; Zbl 0418.60081) and ''Dirichlet forms and Markov processes.'' (1980; Zbl 0422.31007)] generalized this result as follows. If \(X_ t\) is a symmetric diffusion process associated with a Dirichlet space (\({\mathcal E},{\mathcal F})\) and \(u\in {\mathcal F}_{loc}\), then \(u(X_ t)-u(X_ 0)\) can be uniquely decomposed as ''a local martingale \(+\) a continuous additive functional (CAF) locally of zero energy''. In this paper, we prove a partial converse of this result. More precisely, we prove that the set of CAFs locally of zero energy is exhausted by the ones appearing in the Fukushima's decomposition for all \(u\in {\mathcal F}_{loc}\). Some examples of the representation of CAFs of locally zero energy are also discussed.