Natural decomposition of processes and weak Dirichlet processes

A class of stochastic processes, called “weak Dirichlet processes”, is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C^1$-transformations and under absolutely continuous change of measure. If a weak Dirichlet process has finite energy, as defined by Graversen and Rao, its Doob-Meyer type decomposition is unique. The developed methods have been applied to a study of generalized martingale convolutions.