Infinite dimensional weak Dirichlet processes and convolution type processes

Abstract

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process. The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an It{\^o}'s type formula applied to f (t, X(t)) where f : [0, T ] $\times$ H $\rightarrow$ R is a C 0,1 function and X a convolution type processes.