Infinite dimensional weak Dirichlet processes and convolution type processes

TL;DR

This paper extends infinite dimensional calculus by defining weak Dirichlet processes, which decompose into a local martingale and an orthogonal process, aiding in the analysis of stochastic evolution equations and SPDEs.

Contribution

It introduces the concept of weak Dirichlet processes in infinite dimensions, generalizing convolution type processes and providing a framework for stochastic calculus in this setting.

Findings

Decomposition of convolution type processes into martingale and orthogonal parts.

Application of the framework to stochastic evolution equations and SPDEs.

A substitute for Itô's formula for functions of 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.